Line data Source code
1 : /* Copyright (C) 2000 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 :
15 : /********************************************************************/
16 : /** **/
17 : /** LLL Algorithm and close friends **/
18 : /** **/
19 : /********************************************************************/
20 : #include "pari.h"
21 : #include "paripriv.h"
22 :
23 : #define DEBUGLEVEL DEBUGLEVEL_qf
24 :
25 : /********************************************************************/
26 : /** QR Factorization via Householder matrices **/
27 : /********************************************************************/
28 : static int
29 24488090 : no_prec_pb(GEN x)
30 : {
31 24413300 : return (typ(x) != t_REAL || realprec(x) > DEFAULTPREC
32 48901390 : || expo(x) < DEFAULTPREC>>1);
33 : }
34 : /* Find a Householder transformation which, applied to x[k..#x], zeroes
35 : * x[k+1..#x]; fill L = (mu_{i,j}). Return 0 if precision problem [obtained
36 : * a 0 vector], 1 otherwise */
37 : static int
38 24496970 : FindApplyQ(GEN x, GEN L, GEN B, long k, GEN Q, long prec)
39 : {
40 24496970 : long i, lx = lg(x)-1;
41 24496970 : GEN x2, x1, xd = x + (k-1);
42 :
43 24496970 : x1 = gel(xd,1);
44 24496970 : x2 = mpsqr(x1);
45 24495919 : if (k < lx)
46 : {
47 19305568 : long lv = lx - (k-1) + 1;
48 19305568 : GEN beta, Nx, v = cgetg(lv, t_VEC);
49 76237574 : for (i=2; i<lv; i++) {
50 56932411 : x2 = mpadd(x2, mpsqr(gel(xd,i)));
51 56931545 : gel(v,i) = gel(xd,i);
52 : }
53 19305163 : if (!signe(x2)) return 0;
54 19297007 : Nx = gsqrt(x2, prec); if (signe(x1) < 0) setsigne(Nx, -1);
55 19297846 : gel(v,1) = mpadd(x1, Nx);
56 :
57 19297237 : if (!signe(x1))
58 728911 : beta = gtofp(x2, prec); /* make sure typ(beta) != t_INT */
59 : else
60 18568326 : beta = mpadd(x2, mpmul(Nx,x1));
61 19297505 : gel(Q,k) = mkvec2(invr(beta), v);
62 :
63 19297811 : togglesign(Nx);
64 19297565 : gcoeff(L,k,k) = Nx;
65 : }
66 : else
67 5190351 : gcoeff(L,k,k) = gel(x,k);
68 24487916 : gel(B,k) = x2;
69 70161923 : for (i=1; i<k; i++) gcoeff(L,k,i) = gel(x,i);
70 24487916 : return no_prec_pb(x2);
71 : }
72 :
73 : /* apply Householder transformation Q = [beta,v] to r with t_INT/t_REAL
74 : * coefficients, in place: r -= ((0|v).r * beta) v */
75 : static void
76 45683586 : ApplyQ(GEN Q, GEN r)
77 : {
78 45683586 : GEN s, rd, beta = gel(Q,1), v = gel(Q,2);
79 45683586 : long i, l = lg(v), lr = lg(r);
80 :
81 45683586 : rd = r + (lr - l);
82 45683586 : s = mpmul(gel(v,1), gel(rd,1));
83 473198696 : for (i=2; i<l; i++) s = mpadd(s, mpmul(gel(v,i) ,gel(rd,i)));
84 45678785 : s = mpmul(beta, s);
85 519221285 : for (i=1; i<l; i++)
86 473534013 : if (signe(gel(v,i))) gel(rd,i) = mpsub(gel(rd,i), mpmul(s, gel(v,i)));
87 45687272 : }
88 : /* apply Q[1], ..., Q[j-1] to r */
89 : static GEN
90 16823861 : ApplyAllQ(GEN Q, GEN r, long j)
91 : {
92 16823861 : pari_sp av = avma;
93 : long i;
94 16823861 : r = leafcopy(r);
95 62504966 : for (i=1; i<j; i++) ApplyQ(gel(Q,i), r);
96 16821785 : return gerepilecopy(av, r);
97 : }
98 :
99 : /* same, arbitrary coefficients [20% slower for t_REAL at DEFAULTPREC] */
100 : static void
101 22113 : RgC_ApplyQ(GEN Q, GEN r)
102 : {
103 22113 : GEN s, rd, beta = gel(Q,1), v = gel(Q,2);
104 22113 : long i, l = lg(v), lr = lg(r);
105 :
106 22113 : rd = r + (lr - l);
107 22113 : s = gmul(gel(v,1), gel(rd,1));
108 464373 : for (i=2; i<l; i++) s = gadd(s, gmul(gel(v,i) ,gel(rd,i)));
109 22113 : s = gmul(beta, s);
110 486486 : for (i=1; i<l; i++)
111 464373 : if (signe(gel(v,i))) gel(rd,i) = gsub(gel(rd,i), gmul(s, gel(v,i)));
112 22113 : }
113 : static GEN
114 567 : RgC_ApplyAllQ(GEN Q, GEN r, long j)
115 : {
116 567 : pari_sp av = avma;
117 : long i;
118 567 : r = leafcopy(r);
119 22680 : for (i=1; i<j; i++) RgC_ApplyQ(gel(Q,i), r);
120 567 : return gerepilecopy(av, r);
121 : }
122 :
123 : int
124 21 : RgM_QR_init(GEN x, GEN *pB, GEN *pQ, GEN *pL, long prec)
125 : {
126 21 : x = RgM_gtomp(x, prec);
127 21 : return QR_init(x, pB, pQ, pL, prec);
128 : }
129 :
130 : static void
131 35 : check_householder(GEN Q)
132 : {
133 35 : long i, l = lg(Q);
134 35 : if (typ(Q) != t_VEC) pari_err_TYPE("mathouseholder", Q);
135 854 : for (i = 1; i < l; i++)
136 : {
137 826 : GEN q = gel(Q,i), v;
138 826 : if (typ(q) != t_VEC || lg(q) != 3) pari_err_TYPE("mathouseholder", Q);
139 826 : v = gel(q,2);
140 826 : if (typ(v) != t_VEC || lg(v)+i-2 != l) pari_err_TYPE("mathouseholder", Q);
141 : }
142 28 : }
143 :
144 : GEN
145 35 : mathouseholder(GEN Q, GEN x)
146 : {
147 35 : long l = lg(Q);
148 35 : check_householder(Q);
149 28 : switch(typ(x))
150 : {
151 14 : case t_MAT:
152 14 : if (lg(x) == 1) return cgetg(1, t_MAT);
153 14 : if (lgcols(x) != l+1) pari_err_TYPE("mathouseholder", x);
154 574 : pari_APPLY_same(RgC_ApplyAllQ(Q, gel(x,i), l));
155 7 : case t_COL:
156 7 : if (lg(x) == l+1) return RgC_ApplyAllQ(Q, x, l);
157 : }
158 7 : pari_err_TYPE("mathouseholder", x);
159 : return NULL; /* LCOV_EXCL_LINE */
160 : }
161 :
162 : GEN
163 35 : matqr(GEN x, long flag, long prec)
164 : {
165 35 : pari_sp av = avma;
166 : GEN B, Q, L;
167 35 : long n = lg(x)-1;
168 35 : if (typ(x) != t_MAT) pari_err_TYPE("matqr",x);
169 35 : if (!n)
170 : {
171 14 : if (!flag) retmkvec2(cgetg(1,t_MAT),cgetg(1,t_MAT));
172 7 : retmkvec2(cgetg(1,t_VEC),cgetg(1,t_MAT));
173 : }
174 21 : if (n != nbrows(x)) pari_err_DIM("matqr");
175 21 : if (!RgM_QR_init(x, &B,&Q,&L, prec)) pari_err_PREC("matqr");
176 21 : if (!flag) Q = shallowtrans(mathouseholder(Q, matid(n)));
177 21 : return gerepilecopy(av, mkvec2(Q, shallowtrans(L)));
178 : }
179 :
180 : /* compute B = | x[k] |^2, Q = Householder transforms and L = mu_{i,j} */
181 : int
182 7672860 : QR_init(GEN x, GEN *pB, GEN *pQ, GEN *pL, long prec)
183 : {
184 7672860 : long j, k = lg(x)-1;
185 7672860 : GEN B = cgetg(k+1, t_VEC), Q = cgetg(k, t_VEC), L = zeromatcopy(k,k);
186 29957794 : for (j=1; j<=k; j++)
187 : {
188 24496651 : GEN r = gel(x,j);
189 24496651 : if (j > 1) r = ApplyAllQ(Q, r, j);
190 24496933 : if (!FindApplyQ(r, L, B, j, Q, prec)) return 0;
191 : }
192 5461143 : *pB = B; *pQ = Q; *pL = L; return 1;
193 : }
194 : /* x a square t_MAT with t_INT / t_REAL entries and maximal rank. Return
195 : * qfgaussred(x~*x) */
196 : GEN
197 297294 : gaussred_from_QR(GEN x, long prec)
198 : {
199 297294 : long j, k = lg(x)-1;
200 : GEN B, Q, L;
201 297294 : if (!QR_init(x, &B,&Q,&L, prec)) return NULL;
202 1061913 : for (j=1; j<k; j++)
203 : {
204 764610 : GEN m = gel(L,j), invNx = invr(gel(m,j));
205 : long i;
206 764602 : gel(m,j) = gel(B,j);
207 2959918 : for (i=j+1; i<=k; i++) gel(m,i) = mpmul(invNx, gel(m,i));
208 : }
209 297303 : gcoeff(L,j,j) = gel(B,j);
210 297303 : return shallowtrans(L);
211 : }
212 : GEN
213 14259 : R_from_QR(GEN x, long prec)
214 : {
215 : GEN B, Q, L;
216 14259 : if (!QR_init(x, &B,&Q,&L, prec)) return NULL;
217 14245 : return shallowtrans(L);
218 : }
219 :
220 : /********************************************************************/
221 : /** QR Factorization via Gram-Schmidt **/
222 : /********************************************************************/
223 :
224 : /* return Gram-Schmidt orthogonal basis (f) attached to (e), B is the
225 : * vector of the (f_i . f_i)*/
226 : GEN
227 47782 : RgM_gram_schmidt(GEN e, GEN *ptB)
228 : {
229 47782 : long i,j,lx = lg(e);
230 47782 : GEN f = RgM_shallowcopy(e), B, iB;
231 :
232 47782 : B = cgetg(lx, t_VEC);
233 47782 : iB= cgetg(lx, t_VEC);
234 :
235 102220 : for (i=1;i<lx;i++)
236 : {
237 54436 : GEN p1 = NULL;
238 54436 : pari_sp av = avma;
239 116501 : for (j=1; j<i; j++)
240 : {
241 62065 : GEN mu = gmul(RgV_dotproduct(gel(e,i),gel(f,j)), gel(iB,j));
242 62065 : GEN p2 = gmul(mu, gel(f,j));
243 62065 : p1 = p1? gadd(p1,p2): p2;
244 : }
245 54436 : p1 = p1? gerepileupto(av, gsub(gel(e,i), p1)): gel(e,i);
246 54436 : gel(f,i) = p1;
247 54436 : gel(B,i) = RgV_dotsquare(gel(f,i));
248 54436 : gel(iB,i) = ginv(gel(B,i));
249 : }
250 47784 : *ptB = B; return f;
251 : }
252 :
253 : /* B a Z-basis (which the caller should LLL-reduce for efficiency), t a vector.
254 : * Apply Babai's nearest plane algorithm to (B,t) */
255 : GEN
256 47782 : RgM_Babai(GEN B, GEN t)
257 : {
258 47782 : GEN C, N, G = RgM_gram_schmidt(B, &N), b = t;
259 47784 : long j, n = lg(B)-1;
260 :
261 47784 : C = cgetg(n+1,t_COL);
262 102220 : for (j = n; j > 0; j--)
263 : {
264 54436 : GEN c = gdiv( RgV_dotproduct(b, gel(G,j)), gel(N,j) );
265 : long e;
266 54436 : c = grndtoi(c,&e);
267 54436 : if (e >= 0) return NULL;
268 54436 : if (signe(c)) b = RgC_sub(b, RgC_Rg_mul(gel(B,j), c));
269 54436 : gel(C,j) = c;
270 : }
271 47784 : return C;
272 : }
273 :
274 : /********************************************************************/
275 : /** **/
276 : /** LLL ALGORITHM **/
277 : /** **/
278 : /********************************************************************/
279 : /* Def: a matrix M is said to be -partially reduced- if | m1 +- m2 | >= |m1|
280 : * for any two columns m1 != m2, in M.
281 : *
282 : * Input: an integer matrix mat whose columns are linearly independent. Find
283 : * another matrix T such that mat * T is partially reduced.
284 : *
285 : * Output: mat * T if flag = 0; T if flag != 0,
286 : *
287 : * This routine is designed to quickly reduce lattices in which one row
288 : * is huge compared to the other rows. For example, when searching for a
289 : * polynomial of degree 3 with root a mod N, the four input vectors might
290 : * be the coefficients of
291 : * X^3 - (a^3 mod N), X^2 - (a^2 mod N), X - (a mod N), N.
292 : * All four constant coefficients are O(p) and the rest are O(1). By the
293 : * pigeon-hole principle, the coefficients of the smallest vector in the
294 : * lattice are O(p^(1/4)), hence significant reduction of vector lengths
295 : * can be anticipated.
296 : *
297 : * An improved algorithm would look only at the leading digits of dot*. It
298 : * would use single-precision calculations as much as possible.
299 : *
300 : * Original code: Peter Montgomery (1994) */
301 : static GEN
302 35 : lllintpartialall(GEN m, long flag)
303 : {
304 35 : const long ncol = lg(m)-1;
305 35 : const pari_sp av = avma;
306 : GEN tm1, tm2, mid;
307 :
308 35 : if (ncol <= 1) return flag? matid(ncol): gcopy(m);
309 :
310 14 : tm1 = flag? matid(ncol): NULL;
311 : {
312 14 : const pari_sp av2 = avma;
313 14 : GEN dot11 = ZV_dotsquare(gel(m,1));
314 14 : GEN dot22 = ZV_dotsquare(gel(m,2));
315 14 : GEN dot12 = ZV_dotproduct(gel(m,1), gel(m,2));
316 14 : GEN tm = matid(2); /* For first two columns only */
317 :
318 14 : int progress = 0;
319 14 : long npass2 = 0;
320 :
321 : /* Row reduce the first two columns of m. Our best result so far is
322 : * (first two columns of m)*tm.
323 : *
324 : * Initially tm = 2 x 2 identity matrix.
325 : * Inner products of the reduced matrix are in dot11, dot12, dot22. */
326 49 : while (npass2 < 2 || progress)
327 : {
328 42 : GEN dot12new, q = diviiround(dot12, dot22);
329 :
330 35 : npass2++; progress = signe(q);
331 35 : if (progress)
332 : {/* Conceptually replace (v1, v2) by (v1 - q*v2, v2), where v1 and v2
333 : * represent the reduced basis for the first two columns of the matrix.
334 : * We do this by updating tm and the inner products. */
335 21 : togglesign(q);
336 21 : dot12new = addii(dot12, mulii(q, dot22));
337 21 : dot11 = addii(dot11, mulii(q, addii(dot12, dot12new)));
338 21 : dot12 = dot12new;
339 21 : ZC_lincomb1_inplace(gel(tm,1), gel(tm,2), q);
340 : }
341 :
342 : /* Interchange the output vectors v1 and v2. */
343 35 : swap(dot11,dot22);
344 35 : swap(gel(tm,1), gel(tm,2));
345 :
346 : /* Occasionally (including final pass) do garbage collection. */
347 35 : if ((npass2 & 0xff) == 0 || !progress)
348 14 : gerepileall(av2, 4, &dot11,&dot12,&dot22,&tm);
349 : } /* while npass2 < 2 || progress */
350 :
351 : {
352 : long i;
353 7 : GEN det12 = subii(mulii(dot11, dot22), sqri(dot12));
354 :
355 7 : mid = cgetg(ncol+1, t_MAT);
356 21 : for (i = 1; i <= 2; i++)
357 : {
358 14 : GEN tmi = gel(tm,i);
359 14 : if (tm1)
360 : {
361 14 : GEN tm1i = gel(tm1,i);
362 14 : gel(tm1i,1) = gel(tmi,1);
363 14 : gel(tm1i,2) = gel(tmi,2);
364 : }
365 14 : gel(mid,i) = ZC_lincomb(gel(tmi,1),gel(tmi,2), gel(m,1),gel(m,2));
366 : }
367 42 : for (i = 3; i <= ncol; i++)
368 : {
369 35 : GEN c = gel(m,i);
370 35 : GEN dot1i = ZV_dotproduct(gel(mid,1), c);
371 35 : GEN dot2i = ZV_dotproduct(gel(mid,2), c);
372 : /* ( dot11 dot12 ) (q1) ( dot1i )
373 : * ( dot12 dot22 ) (q2) = ( dot2i )
374 : *
375 : * Round -q1 and -q2 to nearest integer. Then compute
376 : * c - q1*mid[1] - q2*mid[2].
377 : * This will be approximately orthogonal to the first two vectors in
378 : * the new basis. */
379 35 : GEN q1neg = subii(mulii(dot12, dot2i), mulii(dot22, dot1i));
380 35 : GEN q2neg = subii(mulii(dot12, dot1i), mulii(dot11, dot2i));
381 :
382 35 : q1neg = diviiround(q1neg, det12);
383 35 : q2neg = diviiround(q2neg, det12);
384 35 : if (tm1)
385 : {
386 35 : gcoeff(tm1,1,i) = addii(mulii(q1neg, gcoeff(tm,1,1)),
387 35 : mulii(q2neg, gcoeff(tm,1,2)));
388 35 : gcoeff(tm1,2,i) = addii(mulii(q1neg, gcoeff(tm,2,1)),
389 35 : mulii(q2neg, gcoeff(tm,2,2)));
390 : }
391 35 : gel(mid,i) = ZC_add(c, ZC_lincomb(q1neg,q2neg, gel(mid,1),gel(mid,2)));
392 : } /* for i */
393 : } /* local block */
394 : }
395 7 : if (DEBUGLEVEL>6)
396 : {
397 0 : if (tm1) err_printf("tm1 = %Ps",tm1);
398 0 : err_printf("mid = %Ps",mid); /* = m * tm1 */
399 : }
400 7 : gerepileall(av, tm1? 2: 1, &mid, &tm1);
401 : {
402 : /* For each pair of column vectors v and w in mid * tm2,
403 : * try to replace (v, w) by (v, v - q*w) for some q.
404 : * We compute all inner products and check them repeatedly. */
405 7 : const pari_sp av3 = avma;
406 7 : long i, j, npass = 0, e = LONG_MAX;
407 7 : GEN dot = cgetg(ncol+1, t_MAT); /* scalar products */
408 :
409 7 : tm2 = matid(ncol);
410 56 : for (i=1; i <= ncol; i++)
411 : {
412 49 : gel(dot,i) = cgetg(ncol+1,t_COL);
413 245 : for (j=1; j <= i; j++)
414 196 : gcoeff(dot,j,i) = gcoeff(dot,i,j) = ZV_dotproduct(gel(mid,i),gel(mid,j));
415 : }
416 : for(;;)
417 35 : {
418 42 : long reductions = 0, olde = e;
419 336 : for (i=1; i <= ncol; i++)
420 : {
421 : long ijdif;
422 2058 : for (ijdif=1; ijdif < ncol; ijdif++)
423 : {
424 : long d, k1, k2;
425 : GEN codi, q;
426 :
427 1764 : j = i + ijdif; if (j > ncol) j -= ncol;
428 : /* let k1, resp. k2, index of larger, resp. smaller, column */
429 1764 : if (cmpii(gcoeff(dot,i,i), gcoeff(dot,j,j)) > 0) { k1 = i; k2 = j; }
430 1022 : else { k1 = j; k2 = i; }
431 1764 : codi = gcoeff(dot,k2,k2);
432 1764 : q = signe(codi)? diviiround(gcoeff(dot,k1,k2), codi): gen_0;
433 1764 : if (!signe(q)) continue;
434 :
435 : /* Try to subtract a multiple of column k2 from column k1. */
436 700 : reductions++; togglesign_safe(&q);
437 700 : ZC_lincomb1_inplace(gel(tm2,k1), gel(tm2,k2), q);
438 700 : ZC_lincomb1_inplace(gel(dot,k1), gel(dot,k2), q);
439 700 : gcoeff(dot,k1,k1) = addii(gcoeff(dot,k1,k1),
440 700 : mulii(q, gcoeff(dot,k2,k1)));
441 5600 : for (d = 1; d <= ncol; d++) gcoeff(dot,k1,d) = gcoeff(dot,d,k1);
442 : } /* for ijdif */
443 294 : if (gc_needed(av3,2))
444 : {
445 0 : if(DEBUGMEM>1) pari_warn(warnmem,"lllintpartialall");
446 0 : gerepileall(av3, 2, &dot,&tm2);
447 : }
448 : } /* for i */
449 42 : if (!reductions) break;
450 35 : e = 0;
451 280 : for (i = 1; i <= ncol; i++) e += expi( gcoeff(dot,i,i) );
452 35 : if (e == olde) break;
453 35 : if (DEBUGLEVEL>6)
454 : {
455 0 : npass++;
456 0 : err_printf("npass = %ld, red. last time = %ld, log_2(det) ~ %ld\n\n",
457 : npass, reductions, e);
458 : }
459 : } /* for(;;)*/
460 :
461 : /* Sort columns so smallest comes first in m * tm1 * tm2.
462 : * Use insertion sort. */
463 49 : for (i = 1; i < ncol; i++)
464 : {
465 42 : long j, s = i;
466 :
467 189 : for (j = i+1; j <= ncol; j++)
468 147 : if (cmpii(gcoeff(dot,s,s),gcoeff(dot,j,j)) > 0) s = j;
469 42 : if (i != s)
470 : { /* Exchange with proper column; only the diagonal of dot is updated */
471 28 : swap(gel(tm2,i), gel(tm2,s));
472 28 : swap(gcoeff(dot,i,i), gcoeff(dot,s,s));
473 : }
474 : }
475 : } /* local block */
476 7 : return gerepileupto(av, ZM_mul(tm1? tm1: mid, tm2));
477 : }
478 :
479 : GEN
480 35 : lllintpartial(GEN mat) { return lllintpartialall(mat,1); }
481 :
482 : GEN
483 0 : lllintpartial_inplace(GEN mat) { return lllintpartialall(mat,0); }
484 :
485 : /********************************************************************/
486 : /** **/
487 : /** COPPERSMITH ALGORITHM **/
488 : /** Finding small roots of univariate equations. **/
489 : /** **/
490 : /********************************************************************/
491 :
492 : static int
493 882 : check(double b, double x, double rho, long d, long dim, long delta, long t)
494 : {
495 882 : double cond = delta * (d * (delta+1) - 2*b*dim + rho * (delta-1 + 2*t))
496 882 : + x*dim*(dim - 1);
497 882 : if (DEBUGLEVEL >= 4)
498 0 : err_printf("delta = %d, t = %d (%.1lf)\n", delta, t, cond);
499 882 : return (cond <= 0);
500 : }
501 :
502 : static void
503 21 : choose_params(GEN P, GEN N, GEN X, GEN B, long *pdelta, long *pt)
504 : {
505 21 : long d = degpol(P), dim;
506 21 : GEN P0 = leading_coeff(P);
507 21 : double logN = gtodouble(glog(N, DEFAULTPREC)), x, b, rho;
508 21 : x = gtodouble(glog(X, DEFAULTPREC)) / logN;
509 21 : b = B? gtodouble(glog(B, DEFAULTPREC)) / logN: 1.;
510 21 : if (x * d >= b * b) pari_err_OVERFLOW("zncoppersmith [bound too large]");
511 : /* TODO : remove P0 completely */
512 14 : rho = is_pm1(P0)? 0: gtodouble(glog(P0, DEFAULTPREC)) / logN;
513 :
514 : /* Enumerate (delta,t) by increasing lattice dimension */
515 14 : for(dim = d + 1;; dim++)
516 161 : {
517 : long delta, t; /* dim = d*delta + t in the loop */
518 1043 : for (delta = 0, t = dim; t >= 0; delta++, t -= d)
519 882 : if (check(b,x,rho,d,dim,delta,t)) { *pdelta = delta; *pt = t; return; }
520 : }
521 : }
522 :
523 : static int
524 14021 : sol_OK(GEN x, GEN N, GEN B)
525 14021 : { return B? (cmpii(gcdii(x,N),B) >= 0): dvdii(x,N); }
526 : /* deg(P) > 0, x >= 0. Find all j such that gcd(P(j), N) >= B, |j| <= x */
527 : static GEN
528 7 : do_exhaustive(GEN P, GEN N, long x, GEN B)
529 : {
530 7 : GEN Pe, Po, sol = vecsmalltrunc_init(2*x + 2);
531 : pari_sp av;
532 : long j;
533 7 : RgX_even_odd(P, &Pe,&Po); av = avma;
534 7 : if (sol_OK(gel(P,2), N,B)) vecsmalltrunc_append(sol, 0);
535 7007 : for (j = 1; j <= x; j++, set_avma(av))
536 : {
537 7000 : GEN j2 = sqru(j), E = FpX_eval(Pe,j2,N), O = FpX_eval(Po,j2,N);
538 7000 : if (sol_OK(addmuliu(E,O,j), N,B)) vecsmalltrunc_append(sol, j);
539 7000 : if (sol_OK(submuliu(E,O,j), N,B)) vecsmalltrunc_append(sol,-j);
540 : }
541 7 : vecsmall_sort(sol); return zv_to_ZV(sol);
542 : }
543 :
544 : /* General Coppersmith, look for a root x0 <= p, p >= B, p | N, |x0| <= X.
545 : * B = N coded as NULL */
546 : GEN
547 35 : zncoppersmith(GEN P, GEN N, GEN X, GEN B)
548 : {
549 : GEN Q, R, N0, M, sh, short_pol, *Xpowers, sol, nsp, cP, Z;
550 35 : long delta, i, j, row, d, l, t, dim, bnd = 10;
551 35 : const ulong X_SMALL = 1000;
552 35 : pari_sp av = avma;
553 :
554 35 : if (typ(P) != t_POL || !RgX_is_ZX(P)) pari_err_TYPE("zncoppersmith",P);
555 28 : if (typ(N) != t_INT) pari_err_TYPE("zncoppersmith",N);
556 28 : if (typ(X) != t_INT) {
557 7 : X = gfloor(X);
558 7 : if (typ(X) != t_INT) pari_err_TYPE("zncoppersmith",X);
559 : }
560 28 : if (signe(X) < 0) pari_err_DOMAIN("zncoppersmith", "X", "<", gen_0, X);
561 28 : P = FpX_red(P, N); d = degpol(P);
562 28 : if (d == 0) { set_avma(av); return cgetg(1, t_VEC); }
563 28 : if (d < 0) pari_err_ROOTS0("zncoppersmith");
564 28 : if (B && typ(B) != t_INT) B = gceil(B);
565 28 : if (abscmpiu(X, X_SMALL) <= 0)
566 7 : return gerepileupto(av, do_exhaustive(P, N, itos(X), B));
567 :
568 21 : if (B && equalii(B,N)) B = NULL;
569 21 : if (B) bnd = 1; /* bnd-hack is only for the case B = N */
570 21 : cP = gel(P,d+2);
571 21 : if (!gequal1(cP))
572 : {
573 : GEN r, z;
574 14 : gel(P,d+2) = cP = bezout(cP, N, &z, &r);
575 35 : for (j = 0; j < d; j++) gel(P,j+2) = Fp_mul(gel(P,j+2), z, N);
576 14 : if (!is_pm1(cP))
577 : {
578 7 : P = Q_primitive_part(P, &cP);
579 7 : if (cP) { N = diviiexact(N,cP); B = gceil(gdiv(B, cP)); }
580 : }
581 : }
582 21 : if (DEBUGLEVEL >= 2) err_printf("Modified P: %Ps\n", P);
583 :
584 21 : choose_params(P, N, X, B, &delta, &t);
585 14 : if (DEBUGLEVEL >= 2)
586 0 : err_printf("Init: trying delta = %d, t = %d\n", delta, t);
587 : for(;;)
588 : {
589 14 : dim = d * delta + t;
590 : /* TODO: In case of failure do not recompute the full vector */
591 14 : Xpowers = (GEN*)new_chunk(dim + 1);
592 14 : Xpowers[0] = gen_1;
593 217 : for (j = 1; j <= dim; j++) Xpowers[j] = mulii(Xpowers[j-1], X);
594 :
595 : /* TODO: in case of failure, use the part of the matrix already computed */
596 14 : M = zeromatcopy(dim,dim);
597 :
598 : /* Rows of M correspond to the polynomials
599 : * N^delta, N^delta Xi, ... N^delta (Xi)^d-1,
600 : * N^(delta-1)P(Xi), N^(delta-1)XiP(Xi), ... N^(delta-1)P(Xi)(Xi)^d-1,
601 : * ...
602 : * P(Xi)^delta, XiP(Xi)^delta, ..., P(Xi)^delta(Xi)^t-1 */
603 42 : for (j = 1; j <= d; j++) gcoeff(M, j, j) = gel(Xpowers,j-1);
604 :
605 : /* P-part */
606 14 : if (delta) row = d + 1; else row = 0;
607 :
608 14 : Q = P;
609 70 : for (i = 1; i < delta; i++)
610 : {
611 182 : for (j = 0; j < d; j++,row++)
612 1239 : for (l = j + 1; l <= row; l++)
613 1113 : gcoeff(M, l, row) = mulii(Xpowers[l-1], gel(Q,l-j+1));
614 56 : Q = ZX_mul(Q, P);
615 : }
616 63 : for (j = 0; j < t; row++, j++)
617 490 : for (l = j + 1; l <= row; l++)
618 441 : gcoeff(M, l, row) = mulii(Xpowers[l-1], gel(Q,l-j+1));
619 :
620 : /* N-part */
621 14 : row = dim - t; N0 = N;
622 84 : while (row >= 1)
623 : {
624 224 : for (j = 0; j < d; j++,row--)
625 1421 : for (l = 1; l <= row; l++)
626 1267 : gcoeff(M, l, row) = mulii(gmael(M, row, l), N0);
627 70 : if (row >= 1) N0 = mulii(N0, N);
628 : }
629 : /* Z is the upper bound for the L^1 norm of the polynomial,
630 : ie. N^delta if B = N, B^delta otherwise */
631 14 : if (B) Z = powiu(B, delta); else Z = N0;
632 :
633 14 : if (DEBUGLEVEL >= 2)
634 : {
635 0 : if (DEBUGLEVEL >= 6) err_printf("Matrix to be reduced:\n%Ps\n", M);
636 0 : err_printf("Entering LLL\nbitsize bound: %ld\n", expi(Z));
637 0 : err_printf("expected shvector bitsize: %ld\n", expi(ZM_det_triangular(M))/dim);
638 : }
639 :
640 14 : sh = ZM_lll(M, 0.75, LLL_INPLACE);
641 : /* Take the first vector if it is non constant */
642 14 : short_pol = gel(sh,1);
643 14 : if (ZV_isscalar(short_pol)) short_pol = gel(sh, 2);
644 :
645 14 : nsp = gen_0;
646 217 : for (j = 1; j <= dim; j++) nsp = addii(nsp, absi_shallow(gel(short_pol,j)));
647 :
648 14 : if (DEBUGLEVEL >= 2)
649 : {
650 0 : err_printf("Candidate: %Ps\n", short_pol);
651 0 : err_printf("bitsize Norm: %ld\n", expi(nsp));
652 0 : err_printf("bitsize bound: %ld\n", expi(mului(bnd, Z)));
653 : }
654 14 : if (cmpii(nsp, mului(bnd, Z)) < 0) break; /* SUCCESS */
655 :
656 : /* Failed with the precomputed or supplied value */
657 0 : if (++t == d) { delta++; t = 1; }
658 0 : if (DEBUGLEVEL >= 2)
659 0 : err_printf("Increasing dim, delta = %d t = %d\n", delta, t);
660 : }
661 14 : bnd = itos(divii(nsp, Z)) + 1;
662 :
663 14 : while (!signe(gel(short_pol,dim))) dim--;
664 :
665 14 : R = cgetg(dim + 2, t_POL); R[1] = P[1];
666 217 : for (j = 1; j <= dim; j++)
667 203 : gel(R,j+1) = diviiexact(gel(short_pol,j), Xpowers[j-1]);
668 14 : gel(R,2) = subii(gel(R,2), mului(bnd - 1, N0));
669 :
670 14 : sol = cgetg(1, t_VEC);
671 84 : for (i = -bnd + 1; i < bnd; i++)
672 : {
673 70 : GEN r = nfrootsQ(R);
674 70 : if (DEBUGLEVEL >= 2) err_printf("Roots: %Ps\n", r);
675 91 : for (j = 1; j < lg(r); j++)
676 : {
677 21 : GEN z = gel(r,j);
678 21 : if (typ(z) == t_INT && sol_OK(FpX_eval(P,z,N), N,B))
679 14 : sol = shallowconcat(sol, z);
680 : }
681 70 : if (i < bnd) gel(R,2) = addii(gel(R,2), Z);
682 : }
683 14 : return gerepileupto(av, ZV_sort_uniq(sol));
684 : }
685 :
686 : /********************************************************************/
687 : /** **/
688 : /** LINEAR & ALGEBRAIC DEPENDENCE **/
689 : /** **/
690 : /********************************************************************/
691 :
692 : static int
693 1634 : real_indep(GEN re, GEN im, long bit)
694 : {
695 1634 : GEN d = gsub(gmul(gel(re,1),gel(im,2)), gmul(gel(re,2),gel(im,1)));
696 1634 : return (!gequal0(d) && gexpo(d) > - bit);
697 : }
698 :
699 : GEN
700 8813 : lindepfull_bit(GEN x, long bit)
701 : {
702 8813 : long lx = lg(x), ly, i, j;
703 : GEN re, im, M;
704 :
705 8813 : if (! is_vec_t(typ(x))) pari_err_TYPE("lindep2",x);
706 8813 : if (lx <= 2)
707 : {
708 21 : if (lx == 2 && gequal0(x)) return matid(1);
709 14 : return NULL;
710 : }
711 8792 : re = real_i(x);
712 8792 : im = imag_i(x);
713 : /* independent over R ? */
714 8792 : if (lx == 3 && real_indep(re,im,bit)) return NULL;
715 8778 : if (gequal0(im)) im = NULL;
716 8778 : ly = im? lx+2: lx+1;
717 8778 : M = cgetg(lx,t_MAT);
718 41234 : for (i=1; i<lx; i++)
719 : {
720 32456 : GEN c = cgetg(ly,t_COL); gel(M,i) = c;
721 170460 : for (j=1; j<lx; j++) gel(c,j) = gen_0;
722 32456 : gel(c,i) = gen_1;
723 32456 : gel(c,lx) = gtrunc2n(gel(re,i), bit);
724 32456 : if (im) gel(c,lx+1) = gtrunc2n(gel(im,i), bit);
725 : }
726 8778 : return ZM_lll(M, 0.99, LLL_INPLACE);
727 : }
728 : GEN
729 3311 : lindep_bit(GEN x, long bit)
730 : {
731 3311 : pari_sp av = avma;
732 3311 : GEN v, M = lindepfull_bit(x,bit);
733 3311 : if (!M) { set_avma(av); return cgetg(1, t_COL); }
734 3283 : v = gel(M,1); setlg(v, lg(M));
735 3283 : return gerepilecopy(av, v);
736 : }
737 : /* deprecated */
738 : GEN
739 112 : lindep2(GEN x, long dig)
740 : {
741 : long bit;
742 112 : if (dig < 0) pari_err_DOMAIN("lindep2", "accuracy", "<", gen_0, stoi(dig));
743 112 : if (dig) bit = (long) (dig/LOG10_2);
744 : else
745 : {
746 98 : bit = gprecision(x);
747 98 : if (!bit)
748 : {
749 35 : x = Q_primpart(x); /* left on stack */
750 35 : bit = 32 + gexpo(x);
751 : }
752 : else
753 63 : bit = (long)prec2nbits_mul(bit, 0.8);
754 : }
755 112 : return lindep_bit(x, bit);
756 : }
757 :
758 : /* x is a vector of elts of a p-adic field */
759 : GEN
760 28 : lindep_padic(GEN x)
761 : {
762 28 : long i, j, prec = LONG_MAX, nx = lg(x)-1, v;
763 28 : pari_sp av = avma;
764 28 : GEN p = NULL, pn, m, a;
765 :
766 28 : if (nx < 2) return cgetg(1,t_COL);
767 147 : for (i=1; i<=nx; i++)
768 : {
769 119 : GEN c = gel(x,i), q;
770 119 : if (typ(c) != t_PADIC) continue;
771 :
772 91 : j = precp(c); if (j < prec) prec = j;
773 91 : q = gel(c,2);
774 91 : if (!p) p = q; else if (!equalii(p, q)) pari_err_MODULUS("lindep_padic", p, q);
775 : }
776 28 : if (!p) pari_err_TYPE("lindep_padic [not a p-adic vector]",x);
777 28 : v = gvaluation(x,p); pn = powiu(p,prec);
778 28 : if (v) x = gmul(x, powis(p, -v));
779 28 : x = RgV_to_FpV(x, pn);
780 :
781 28 : a = negi(gel(x,1));
782 28 : m = cgetg(nx,t_MAT);
783 119 : for (i=1; i<nx; i++)
784 : {
785 91 : GEN c = zerocol(nx);
786 91 : gel(c,1+i) = a;
787 91 : gel(c,1) = gel(x,i+1);
788 91 : gel(m,i) = c;
789 : }
790 28 : m = ZM_lll(ZM_hnfmodid(m, pn), 0.99, LLL_INPLACE);
791 28 : return gerepilecopy(av, gel(m,1));
792 : }
793 : /* x is a vector of t_POL/t_SER */
794 : GEN
795 77 : lindep_Xadic(GEN x)
796 : {
797 77 : long i, prec = LONG_MAX, deg = 0, lx = lg(x), vx, v;
798 77 : pari_sp av = avma;
799 : GEN m;
800 :
801 77 : if (lx == 1) return cgetg(1,t_COL);
802 77 : vx = gvar(x);
803 77 : if (gequal0(x)) return col_ei(lx-1,1);
804 70 : v = gvaluation(x, pol_x(vx));
805 70 : if (!v) x = shallowcopy(x);
806 0 : else if (v > 0) x = gdiv(x, pol_xn(v, vx));
807 0 : else x = gmul(x, pol_xn(-v, vx));
808 : /* all t_SER have valuation >= 0 */
809 735 : for (i=1; i<lx; i++)
810 : {
811 665 : GEN c = gel(x,i);
812 665 : if (gvar(c) != vx) { gel(x,i) = scalarpol_shallow(c, vx); continue; }
813 658 : switch(typ(c))
814 : {
815 231 : case t_POL: deg = maxss(deg, degpol(c)); break;
816 0 : case t_RFRAC: pari_err_TYPE("lindep_Xadic", c);
817 427 : case t_SER:
818 427 : prec = minss(prec, valser(c)+lg(c)-2);
819 427 : gel(x,i) = ser2rfrac_i(c);
820 : }
821 : }
822 70 : if (prec == LONG_MAX) prec = deg+1;
823 70 : m = RgXV_to_RgM(x, prec);
824 70 : return gerepileupto(av, deplin(m));
825 : }
826 : static GEN
827 35 : vec_lindep(GEN x)
828 : {
829 35 : pari_sp av = avma;
830 35 : long i, l = lg(x); /* > 1 */
831 35 : long t = typ(gel(x,1)), h = lg(gel(x,1));
832 35 : GEN m = cgetg(l, t_MAT);
833 126 : for (i = 1; i < l; i++)
834 : {
835 98 : GEN y = gel(x,i);
836 98 : if (lg(y) != h || typ(y) != t) pari_err_TYPE("lindep",x);
837 91 : if (t != t_COL) y = shallowtrans(y); /* Sigh */
838 91 : gel(m,i) = y;
839 : }
840 28 : return gerepileupto(av, deplin(m));
841 : }
842 :
843 : GEN
844 0 : lindep(GEN x) { return lindep2(x, 0); }
845 :
846 : GEN
847 434 : lindep0(GEN x,long bit)
848 : {
849 434 : long i, tx = typ(x);
850 434 : if (tx == t_MAT) return deplin(x);
851 147 : if (! is_vec_t(tx)) pari_err_TYPE("lindep",x);
852 441 : for (i = 1; i < lg(x); i++)
853 357 : switch(typ(gel(x,i)))
854 : {
855 7 : case t_PADIC: return lindep_padic(x);
856 21 : case t_POL:
857 : case t_RFRAC:
858 21 : case t_SER: return lindep_Xadic(x);
859 35 : case t_VEC:
860 35 : case t_COL: return vec_lindep(x);
861 : }
862 84 : return lindep2(x, bit);
863 : }
864 :
865 : GEN
866 77 : algdep0(GEN x, long n, long bit)
867 : {
868 77 : long tx = typ(x), i;
869 : pari_sp av;
870 : GEN y;
871 :
872 77 : if (! is_scalar_t(tx)) pari_err_TYPE("algdep0",x);
873 77 : if (tx == t_POLMOD)
874 : {
875 14 : av = avma; y = minpoly(x, 0);
876 14 : return (degpol(y) > n)? gc_const(av, gen_1): y;
877 : }
878 63 : if (gequal0(x)) return pol_x(0);
879 63 : if (n <= 0)
880 : {
881 14 : if (!n) return gen_1;
882 7 : pari_err_DOMAIN("algdep", "degree", "<", gen_0, stoi(n));
883 : }
884 :
885 49 : av = avma; y = cgetg(n+2,t_COL);
886 49 : gel(y,1) = gen_1;
887 49 : gel(y,2) = x; /* n >= 1 */
888 210 : for (i=3; i<=n+1; i++) gel(y,i) = gmul(gel(y,i-1),x);
889 49 : if (typ(x) == t_PADIC)
890 21 : y = lindep_padic(y);
891 : else
892 28 : y = lindep2(y, bit);
893 49 : if (lg(y) == 1) pari_err(e_DOMAIN,"algdep", "degree(x)",">", stoi(n), x);
894 49 : y = RgV_to_RgX(y, 0);
895 49 : if (signe(leading_coeff(y)) > 0) return gerepilecopy(av, y);
896 14 : return gerepileupto(av, ZX_neg(y));
897 : }
898 :
899 : GEN
900 0 : algdep(GEN x, long n)
901 : {
902 0 : return algdep0(x,n,0);
903 : }
904 :
905 : static GEN
906 56 : sertomat(GEN S, long p, long r, long vy)
907 : {
908 : long n, m;
909 56 : GEN v = cgetg(r*p+1, t_VEC); /* v[r*n+m+1] = s^n * y^m */
910 : /* n = 0 */
911 245 : for (m = 0; m < r; m++) gel(v, m+1) = pol_xn(m, vy);
912 175 : for(n=1; n < p; n++)
913 546 : for (m = 0; m < r; m++)
914 : {
915 427 : GEN c = gel(S,n);
916 427 : if (m)
917 : {
918 308 : c = shallowcopy(c);
919 308 : setvalser(c, valser(c) + m);
920 : }
921 427 : gel(v, r*n + m + 1) = c;
922 : }
923 56 : return v;
924 : }
925 :
926 : GEN
927 42 : seralgdep(GEN s, long p, long r)
928 : {
929 42 : pari_sp av = avma;
930 : long vy, i, n, prec;
931 : GEN S, v, D;
932 :
933 42 : if (typ(s) != t_SER) pari_err_TYPE("seralgdep",s);
934 42 : if (p <= 0) pari_err_DOMAIN("seralgdep", "p", "<=", gen_0, stoi(p));
935 42 : if (r < 0) pari_err_DOMAIN("seralgdep", "r", "<", gen_0, stoi(r));
936 42 : if (is_bigint(addiu(muluu(p, r), 1))) pari_err_OVERFLOW("seralgdep");
937 42 : vy = varn(s);
938 42 : if (!vy) pari_err_PRIORITY("seralgdep", s, ">", 0);
939 42 : r++; p++;
940 42 : prec = valser(s) + lg(s)-2;
941 42 : if (r > prec) r = prec;
942 42 : S = cgetg(p+1, t_VEC); gel(S, 1) = s;
943 119 : for (i = 2; i <= p; i++) gel(S,i) = gmul(gel(S,i-1), s);
944 42 : v = sertomat(S, p, r, vy);
945 42 : D = lindep_Xadic(v);
946 42 : if (lg(D) == 1) { set_avma(av); return gen_0; }
947 35 : v = cgetg(p+1, t_VEC);
948 133 : for (n = 0; n < p; n++)
949 98 : gel(v, n+1) = RgV_to_RgX(vecslice(D, r*n+1, r*n+r), vy);
950 35 : return gerepilecopy(av, RgV_to_RgX(v, 0));
951 : }
952 :
953 : GEN
954 14 : serdiffdep(GEN s, long p, long r)
955 : {
956 14 : pari_sp av = avma;
957 : long vy, i, n, prec;
958 : GEN P, S, v, D;
959 :
960 14 : if (typ(s) != t_SER) pari_err_TYPE("serdiffdep",s);
961 14 : if (p <= 0) pari_err_DOMAIN("serdiffdep", "p", "<=", gen_0, stoi(p));
962 14 : if (r < 0) pari_err_DOMAIN("serdiffdep", "r", "<", gen_0, stoi(r));
963 14 : if (is_bigint(addiu(muluu(p, r), 1))) pari_err_OVERFLOW("serdiffdep");
964 14 : vy = varn(s);
965 14 : if (!vy) pari_err_PRIORITY("serdiffdep", s, ">", 0);
966 14 : r++; p++;
967 14 : prec = valser(s) + lg(s)-2;
968 14 : if (r > prec) r = prec;
969 14 : S = cgetg(p+1, t_VEC); gel(S, 1) = s;
970 56 : for (i = 2; i <= p; i++) gel(S,i) = derivser(gel(S,i-1));
971 14 : v = sertomat(S, p, r, vy);
972 14 : D = lindep_Xadic(v);
973 14 : if (lg(D) == 1) { set_avma(av); return gen_0; }
974 14 : P = RgV_to_RgX(vecslice(D, 1, r), vy);
975 14 : v = cgetg(p, t_VEC);
976 56 : for (n = 1; n < p; n++)
977 42 : gel(v, n) = RgV_to_RgX(vecslice(D, r*n+1, r*n+r), vy);
978 14 : return gerepilecopy(av, mkvec2(RgV_to_RgX(v, 0), gneg(P)));
979 : }
980 :
981 : /* FIXME: could precompute ZM_lll attached to V[2..] */
982 : static GEN
983 5502 : lindepcx(GEN V, long bit)
984 : {
985 5502 : GEN Vr = real_i(V), Vi = imag_i(V);
986 5502 : if (gexpo(Vr) < -bit) V = Vi;
987 5502 : else if (gexpo(Vi) < -bit) V = Vr;
988 5502 : return lindepfull_bit(V, bit);
989 : }
990 : /* c floating point t_REAL or t_COMPLEX, T ZX, recognize in Q[x]/(T).
991 : * V helper vector (containing complex roots of T), MODIFIED */
992 : static GEN
993 5502 : cx_bestapprnf(GEN c, GEN T, GEN V, long bit)
994 : {
995 5502 : GEN M, a, v = NULL;
996 : long i, l;
997 5502 : gel(V,1) = gneg(c); M = lindepcx(V, bit);
998 5502 : if (!M) pari_err(e_MISC, "cannot rationalize coeff in bestapprnf");
999 5502 : l = lg(M); a = NULL;
1000 5502 : for (i = 1; i < l; i ++) { v = gel(M,i); a = gel(v,1); if (signe(a)) break; }
1001 5502 : v = RgC_Rg_div(vecslice(v, 2, lg(M)-1), a);
1002 5502 : if (!T) return gel(v,1);
1003 4830 : v = RgV_to_RgX(v, varn(T)); l = lg(v);
1004 4830 : if (l == 2) return gen_0;
1005 4165 : if (l == 3) return gel(v,2);
1006 3668 : return mkpolmod(v, T);
1007 : }
1008 : static GEN
1009 8246 : bestapprnf_i(GEN x, GEN T, GEN V, long bit)
1010 : {
1011 8246 : long i, l, tx = typ(x);
1012 : GEN z;
1013 8246 : switch (tx)
1014 : {
1015 833 : case t_INT: case t_FRAC: return x;
1016 5502 : case t_REAL: case t_COMPLEX: return cx_bestapprnf(x, T, V, bit);
1017 0 : case t_POLMOD: if (RgX_equal(gel(x,1),T)) return x;
1018 0 : break;
1019 1911 : case t_POL: case t_SER: case t_VEC: case t_COL: case t_MAT:
1020 1911 : l = lg(x); z = cgetg(l, tx);
1021 3437 : for (i = 1; i < lontyp[tx]; i++) z[i] = x[i];
1022 8211 : for (; i < l; i++) gel(z,i) = bestapprnf_i(gel(x,i), T, V, bit);
1023 1911 : return z;
1024 : }
1025 0 : pari_err_TYPE("mfcxtoQ", x);
1026 : return NULL;/*LCOV_EXCL_LINE*/
1027 : }
1028 :
1029 : GEN
1030 1946 : bestapprnf(GEN x, GEN T, GEN roT, long prec)
1031 : {
1032 1946 : pari_sp av = avma;
1033 1946 : long tx = typ(x), dT = 1, bit;
1034 : GEN V;
1035 :
1036 1946 : if (T)
1037 : {
1038 1610 : if (typ(T) != t_POL) T = nf_get_pol(checknf(T));
1039 1610 : else if (!RgX_is_ZX(T)) pari_err_TYPE("bestapprnf", T);
1040 1610 : dT = degpol(T);
1041 : }
1042 1946 : if (is_rational_t(tx)) return gcopy(x);
1043 1946 : if (tx == t_POLMOD)
1044 : {
1045 0 : if (!T || !RgX_equal(T, gel(x,1))) pari_err_TYPE("bestapprnf",x);
1046 0 : return gcopy(x);
1047 : }
1048 :
1049 1946 : if (roT)
1050 : {
1051 644 : long l = gprecision(roT);
1052 644 : switch(typ(roT))
1053 : {
1054 644 : case t_INT: case t_FRAC: case t_REAL: case t_COMPLEX: break;
1055 0 : default: pari_err_TYPE("bestapprnf", roT);
1056 : }
1057 644 : if (prec < l) prec = l;
1058 : }
1059 1302 : else if (!T)
1060 336 : roT = gen_1;
1061 : else
1062 : {
1063 966 : long n = poliscyclo(T); /* cyclotomic is an important special case */
1064 966 : roT = n? rootsof1u_cx(n,prec): gel(QX_complex_roots(T,prec), 1);
1065 : }
1066 1946 : V = vec_prepend(gpowers(roT, dT-1), NULL);
1067 1946 : bit = prec2nbits_mul(prec, 0.8);
1068 1946 : return gerepilecopy(av, bestapprnf_i(x, T, V, bit));
1069 : }
1070 :
1071 : /********************************************************************/
1072 : /** **/
1073 : /** MINIM **/
1074 : /** **/
1075 : /********************************************************************/
1076 : void
1077 121651 : minim_alloc(long n, double ***q, GEN *x, double **y, double **z, double **v)
1078 : {
1079 : long i, s;
1080 :
1081 121651 : *x = cgetg(n, t_VECSMALL);
1082 121651 : *q = (double**) new_chunk(n);
1083 121651 : s = n * sizeof(double);
1084 121651 : *y = (double*) stack_malloc_align(s, sizeof(double));
1085 121651 : *z = (double*) stack_malloc_align(s, sizeof(double));
1086 121651 : *v = (double*) stack_malloc_align(s, sizeof(double));
1087 525546 : for (i=1; i<n; i++) (*q)[i] = (double*) stack_malloc_align(s, sizeof(double));
1088 121651 : }
1089 :
1090 : static void
1091 70 : cvp_alloc(long n, double ***q, GEN *x, double **y, double **z, double **v, double **t, double **tpre)
1092 : {
1093 : long i, s;
1094 :
1095 70 : *x = cgetg(n, t_VECSMALL);
1096 70 : *q = (double**) new_chunk(n);
1097 70 : s = n * sizeof(double);
1098 70 : *y = (double*) stack_malloc_align(s, sizeof(double));
1099 70 : *z = (double*) stack_malloc_align(s, sizeof(double));
1100 70 : *v = (double*) stack_malloc_align(s, sizeof(double));
1101 70 : *t = (double*) stack_malloc_align(s, sizeof(double));
1102 70 : *tpre = (double*) stack_malloc_align(s, sizeof(double));
1103 392 : for (i=1; i<n; i++) (*q)[i] = (double*) stack_malloc_align(s, sizeof(double));
1104 70 : }
1105 :
1106 : static GEN
1107 245868 : ZC_canon(GEN V)
1108 : {
1109 245868 : long l = lg(V), j;
1110 571655 : for (j = 1; j < l && signe(gel(V,j)) == 0; ++j);
1111 245868 : return (j < l && signe(gel(V,j)) < 0)? ZC_neg(V): V;
1112 : }
1113 :
1114 : static GEN
1115 5502 : ZM_zc_mul_canon(GEN u, GEN x)
1116 : {
1117 5502 : return ZC_canon(ZM_zc_mul(u,x));
1118 : }
1119 :
1120 : static GEN
1121 240366 : ZM_zc_mul_canon_zm(GEN u, GEN x)
1122 : {
1123 240366 : pari_sp av = avma;
1124 240366 : GEN M = ZV_to_zv(ZC_canon(ZM_zc_mul(u,x)));
1125 240366 : return gerepileupto(av, M);
1126 : }
1127 :
1128 : struct qfvec
1129 : {
1130 : GEN a, r, u;
1131 : };
1132 :
1133 : static void
1134 0 : err_minim(GEN a)
1135 : {
1136 0 : pari_err_DOMAIN("minim0","form","is not",
1137 : strtoGENstr("positive definite"),a);
1138 0 : }
1139 :
1140 : static GEN
1141 902 : minim_lll(GEN a, GEN *u)
1142 : {
1143 902 : *u = lllgramint(a);
1144 902 : if (lg(*u) != lg(a)) err_minim(a);
1145 902 : return qf_ZM_apply(a,*u);
1146 : }
1147 :
1148 : static void
1149 902 : forqfvec_init_dolll(struct qfvec *qv, GEN *pa, long dolll)
1150 : {
1151 902 : GEN r, u, a = *pa;
1152 902 : if (!dolll) u = NULL;
1153 : else
1154 : {
1155 860 : if (typ(a) != t_MAT || !RgM_is_ZM(a)) pari_err_TYPE("qfminim",a);
1156 860 : a = *pa = minim_lll(a, &u);
1157 : }
1158 902 : qv->a = RgM_gtofp(a, DEFAULTPREC);
1159 902 : r = qfgaussred_positive(qv->a);
1160 902 : if (!r)
1161 : {
1162 0 : r = qfgaussred_positive(a); /* exact computation */
1163 0 : if (!r) err_minim(a);
1164 0 : r = RgM_gtofp(r, DEFAULTPREC);
1165 : }
1166 902 : qv->r = r;
1167 902 : qv->u = u;
1168 902 : }
1169 :
1170 : static void
1171 42 : forqfvec_init(struct qfvec *qv, GEN a)
1172 42 : { forqfvec_init_dolll(qv, &a, 1); }
1173 :
1174 : static void
1175 42 : forqfvec_i(void *E, long (*fun)(void *, GEN, GEN, double), struct qfvec *qv, GEN BORNE)
1176 : {
1177 42 : GEN x, a = qv->a, r = qv->r, u = qv->u;
1178 42 : long n = lg(a)-1, i, j, k;
1179 : double p,BOUND,*v,*y,*z,**q;
1180 42 : const double eps = 1e-10;
1181 42 : if (!BORNE) BORNE = gen_0;
1182 : else
1183 : {
1184 28 : BORNE = gfloor(BORNE);
1185 28 : if (typ(BORNE) != t_INT) pari_err_TYPE("minim0",BORNE);
1186 35 : if (signe(BORNE) <= 0) return;
1187 : }
1188 35 : if (n == 0) return;
1189 28 : minim_alloc(n+1, &q, &x, &y, &z, &v);
1190 98 : for (j=1; j<=n; j++)
1191 : {
1192 70 : v[j] = rtodbl(gcoeff(r,j,j));
1193 133 : for (i=1; i<j; i++) q[i][j] = rtodbl(gcoeff(r,i,j));
1194 : }
1195 :
1196 28 : if (gequal0(BORNE))
1197 : {
1198 : double c;
1199 14 : p = rtodbl(gcoeff(a,1,1));
1200 42 : for (i=2; i<=n; i++) { c = rtodbl(gcoeff(a,i,i)); if (c < p) p = c; }
1201 14 : BORNE = roundr(dbltor(p));
1202 : }
1203 : else
1204 14 : p = gtodouble(BORNE);
1205 28 : BOUND = p * (1 + eps);
1206 28 : if (BOUND > (double)ULONG_MAX || (ulong)BOUND != (ulong)p)
1207 7 : pari_err_PREC("forqfvec");
1208 :
1209 21 : k = n; y[n] = z[n] = 0;
1210 21 : x[n] = (long)sqrt(BOUND/v[n]);
1211 56 : for(;;x[1]--)
1212 : {
1213 : do
1214 : {
1215 140 : if (k>1)
1216 : {
1217 84 : long l = k-1;
1218 84 : z[l] = 0;
1219 245 : for (j=k; j<=n; j++) z[l] += q[l][j]*x[j];
1220 84 : p = (double)x[k] + z[k];
1221 84 : y[l] = y[k] + p*p*v[k];
1222 84 : x[l] = (long)floor(sqrt((BOUND-y[l])/v[l])-z[l]);
1223 84 : k = l;
1224 : }
1225 : for(;;)
1226 : {
1227 189 : p = (double)x[k] + z[k];
1228 189 : if (y[k] + p*p*v[k] <= BOUND) break;
1229 49 : k++; x[k]--;
1230 : }
1231 140 : } while (k > 1);
1232 77 : if (! x[1] && y[1]<=eps) break;
1233 :
1234 56 : p = (double)x[1] + z[1]; p = y[1] + p*p*v[1]; /* norm(x) */
1235 56 : if (fun(E, u, x, p)) break;
1236 : }
1237 : }
1238 :
1239 : void
1240 0 : forqfvec(void *E, long (*fun)(void *, GEN, GEN, double), GEN a, GEN BORNE)
1241 : {
1242 0 : pari_sp av = avma;
1243 : struct qfvec qv;
1244 0 : forqfvec_init(&qv, a);
1245 0 : forqfvec_i(E, fun, &qv, BORNE);
1246 0 : set_avma(av);
1247 0 : }
1248 :
1249 : struct qfvecwrap
1250 : {
1251 : void *E;
1252 : long (*fun)(void *, GEN);
1253 : };
1254 :
1255 : static long
1256 56 : forqfvec_wrap(void *E, GEN u, GEN x, double d)
1257 : {
1258 56 : pari_sp av = avma;
1259 56 : struct qfvecwrap *W = (struct qfvecwrap *) E;
1260 : (void) d;
1261 56 : return gc_long(av, W->fun(W->E, ZM_zc_mul_canon(u, x)));
1262 : }
1263 :
1264 : void
1265 42 : forqfvec1(void *E, long (*fun)(void *, GEN), GEN a, GEN BORNE)
1266 : {
1267 42 : pari_sp av = avma;
1268 : struct qfvecwrap wr;
1269 : struct qfvec qv;
1270 42 : wr.E = E; wr.fun = fun;
1271 42 : forqfvec_init(&qv, a);
1272 42 : forqfvec_i((void*) &wr, forqfvec_wrap, &qv, BORNE);
1273 35 : set_avma(av);
1274 35 : }
1275 :
1276 : void
1277 42 : forqfvec0(GEN a, GEN BORNE, GEN code)
1278 42 : { EXPRVOID_WRAP(code, forqfvec1(EXPR_ARGVOID, a, BORNE)) }
1279 :
1280 : enum { min_ALL = 0, min_FIRST, min_VECSMALL, min_VECSMALL2 };
1281 :
1282 : /* Minimal vectors for the integral definite quadratic form: a.
1283 : * Result u:
1284 : * u[1]= Number of vectors of square norm <= BORNE
1285 : * u[2]= maximum norm found
1286 : * u[3]= list of vectors found (at most STOCKMAX, unless NULL)
1287 : *
1288 : * If BORNE = NULL: Minimal nonzero vectors.
1289 : * flag = min_ALL, as above
1290 : * flag = min_FIRST, exits when first suitable vector is found.
1291 : * flag = min_VECSMALL, return a t_VECSMALL of (half) the number of vectors
1292 : * for each norm
1293 : * flag = min_VECSMALL2, same but count only vectors with even norm, and shift
1294 : * the answer */
1295 : static GEN
1296 847 : minim0_dolll(GEN a, GEN BORNE, GEN STOCKMAX, long flag, long dolll)
1297 : {
1298 : GEN x, u, r, L, gnorme;
1299 847 : long n = lg(a)-1, i, j, k, s, maxrank, sBORNE;
1300 847 : pari_sp av = avma, av1;
1301 : double p,maxnorm,BOUND,*v,*y,*z,**q;
1302 847 : const double eps = 1e-10;
1303 847 : int stockall = 0;
1304 : struct qfvec qv;
1305 :
1306 847 : if (!BORNE)
1307 56 : sBORNE = 0;
1308 : else
1309 : {
1310 791 : BORNE = gfloor(BORNE);
1311 791 : if (typ(BORNE) != t_INT) pari_err_TYPE("minim0",BORNE);
1312 791 : if (is_bigint(BORNE)) pari_err_PREC( "qfminim");
1313 790 : sBORNE = itos(BORNE); set_avma(av);
1314 790 : if (sBORNE < 0) sBORNE = 0;
1315 : }
1316 846 : if (!STOCKMAX)
1317 : {
1318 335 : stockall = 1;
1319 335 : maxrank = 200;
1320 : }
1321 : else
1322 : {
1323 511 : STOCKMAX = gfloor(STOCKMAX);
1324 511 : if (typ(STOCKMAX) != t_INT) pari_err_TYPE("minim0",STOCKMAX);
1325 511 : maxrank = itos(STOCKMAX);
1326 511 : if (maxrank < 0)
1327 0 : pari_err_TYPE("minim0 [negative number of vectors]",STOCKMAX);
1328 : }
1329 :
1330 846 : switch(flag)
1331 : {
1332 462 : case min_VECSMALL:
1333 : case min_VECSMALL2:
1334 462 : if (sBORNE <= 0) return cgetg(1, t_VECSMALL);
1335 434 : L = zero_zv(sBORNE);
1336 434 : if (flag == min_VECSMALL2) sBORNE <<= 1;
1337 434 : if (n == 0) return L;
1338 434 : break;
1339 35 : case min_FIRST:
1340 35 : if (n == 0 || (!sBORNE && BORNE)) return cgetg(1,t_VEC);
1341 21 : L = NULL; /* gcc -Wall */
1342 21 : break;
1343 349 : case min_ALL:
1344 349 : if (n == 0 || (!sBORNE && BORNE))
1345 14 : retmkvec3(gen_0, gen_0, cgetg(1, t_MAT));
1346 335 : L = new_chunk(1+maxrank);
1347 335 : break;
1348 0 : default:
1349 0 : return NULL;
1350 : }
1351 790 : minim_alloc(n+1, &q, &x, &y, &z, &v);
1352 :
1353 790 : forqfvec_init_dolll(&qv, &a, dolll);
1354 790 : av1 = avma;
1355 790 : r = qv.r;
1356 790 : u = qv.u;
1357 5912 : for (j=1; j<=n; j++)
1358 : {
1359 5122 : v[j] = rtodbl(gcoeff(r,j,j));
1360 29579 : for (i=1; i<j; i++) q[i][j] = rtodbl(gcoeff(r,i,j)); /* |.| <= 1/2 */
1361 : }
1362 :
1363 790 : if (sBORNE) maxnorm = 0.;
1364 : else
1365 : {
1366 56 : GEN B = gcoeff(a,1,1);
1367 56 : long t = 1;
1368 616 : for (i=2; i<=n; i++)
1369 : {
1370 560 : GEN c = gcoeff(a,i,i);
1371 560 : if (cmpii(c, B) < 0) { B = c; t = i; }
1372 : }
1373 56 : if (flag == min_FIRST) return gerepilecopy(av, mkvec2(B, gel(u,t)));
1374 49 : maxnorm = -1.; /* don't update maxnorm */
1375 49 : if (is_bigint(B)) return NULL;
1376 48 : sBORNE = itos(B);
1377 : }
1378 782 : BOUND = sBORNE * (1 + eps);
1379 782 : if ((long)BOUND != sBORNE) return NULL;
1380 :
1381 770 : s = 0;
1382 770 : k = n; y[n] = z[n] = 0;
1383 770 : x[n] = (long)sqrt(BOUND/v[n]);
1384 1223264 : for(;;x[1]--)
1385 : {
1386 : do
1387 : {
1388 2245614 : if (k>1)
1389 : {
1390 1022259 : long l = k-1;
1391 1022259 : z[l] = 0;
1392 11756080 : for (j=k; j<=n; j++) z[l] += q[l][j]*x[j];
1393 1022259 : p = (double)x[k] + z[k];
1394 1022259 : y[l] = y[k] + p*p*v[k];
1395 1022259 : x[l] = (long)floor(sqrt((BOUND-y[l])/v[l])-z[l]);
1396 1022259 : k = l;
1397 : }
1398 : for(;;)
1399 : {
1400 3263729 : p = (double)x[k] + z[k];
1401 3263729 : if (y[k] + p*p*v[k] <= BOUND) break;
1402 1018115 : k++; x[k]--;
1403 : }
1404 : }
1405 2245614 : while (k > 1);
1406 1224034 : if (! x[1] && y[1]<=eps) break;
1407 :
1408 1223271 : p = (double)x[1] + z[1]; p = y[1] + p*p*v[1]; /* norm(x) */
1409 1223271 : if (maxnorm >= 0)
1410 : {
1411 1220723 : if (p > maxnorm) maxnorm = p;
1412 : }
1413 : else
1414 : { /* maxnorm < 0 : only look for minimal vectors */
1415 2548 : pari_sp av2 = avma;
1416 2548 : gnorme = roundr(dbltor(p));
1417 2548 : if (cmpis(gnorme, sBORNE) >= 0) set_avma(av2);
1418 : else
1419 : {
1420 14 : sBORNE = itos(gnorme); set_avma(av1);
1421 14 : BOUND = sBORNE * (1+eps);
1422 14 : L = new_chunk(maxrank+1);
1423 14 : s = 0;
1424 : }
1425 : }
1426 1223271 : s++;
1427 :
1428 1223271 : switch(flag)
1429 : {
1430 7 : case min_FIRST:
1431 7 : if (dolll) x = ZM_zc_mul_canon(u,x);
1432 7 : return gerepilecopy(av, mkvec2(roundr(dbltor(p)), x));
1433 :
1434 248241 : case min_ALL:
1435 248241 : if (s > maxrank && stockall) /* overflow */
1436 : {
1437 490 : long maxranknew = maxrank << 1;
1438 490 : GEN Lnew = new_chunk(1 + maxranknew);
1439 344890 : for (i=1; i<=maxrank; i++) Lnew[i] = L[i];
1440 490 : L = Lnew; maxrank = maxranknew;
1441 : }
1442 248241 : if (s<=maxrank) gel(L,s) = leafcopy(x);
1443 248241 : break;
1444 :
1445 39200 : case min_VECSMALL:
1446 39200 : { ulong norm = (ulong)(p + 0.5); L[norm]++; }
1447 39200 : break;
1448 :
1449 935823 : case min_VECSMALL2:
1450 935823 : { ulong norm = (ulong)(p + 0.5); if (!odd(norm)) L[norm>>1]++; }
1451 935823 : break;
1452 :
1453 : }
1454 : }
1455 763 : switch(flag)
1456 : {
1457 7 : case min_FIRST:
1458 7 : set_avma(av); return cgetg(1,t_VEC);
1459 434 : case min_VECSMALL:
1460 : case min_VECSMALL2:
1461 434 : set_avma((pari_sp)L); return L;
1462 : }
1463 322 : r = (maxnorm >= 0) ? roundr(dbltor(maxnorm)): stoi(sBORNE);
1464 322 : k = minss(s,maxrank);
1465 322 : L[0] = evaltyp(t_MAT) | evallg(k + 1);
1466 322 : if (dolll)
1467 246092 : for (j=1; j<=k; j++)
1468 245805 : gel(L,j) = dolll==1 ? ZM_zc_mul_canon(u, gel(L,j))
1469 245805 : : ZM_zc_mul_canon_zm(u, gel(L,j));
1470 322 : return gerepilecopy(av, mkvec3(stoi(s<<1), r, L));
1471 : }
1472 :
1473 : /* Closest vectors for the integral definite quadratic form: a.
1474 : * Code bases on minim0_dolll
1475 : * Result u:
1476 : * u[1]= Number of closest vectors of square distance <= BORNE
1477 : * u[2]= maximum squared distance found
1478 : * u[3]= list of vectors found (at most STOCKMAX, unless NULL)
1479 : *
1480 : * If BORNE = NULL or <= 0.: returns closest vectors.
1481 : * flag = min_ALL, as above
1482 : * flag = min_FIRST, exits when first suitable vector is found.
1483 : */
1484 : static GEN
1485 91 : cvp0_dolll(GEN a, GEN target, GEN BORNE, GEN STOCKMAX, long flag, long dolll)
1486 : {
1487 : GEN x, u, r, L;
1488 : GEN uinv, tv;
1489 : GEN pd;
1490 91 : long n = lg(a)-1, nt = lg(target)-1, i, j, k, s, maxrank;
1491 91 : pari_sp av = avma, av1;
1492 : double p,maxnorm,BOUND,*v,*y,*z,*tt,**q, *tpre, sBORNE;
1493 91 : const double eps = 1e-10;
1494 91 : int stockall = 0;
1495 : struct qfvec qv;
1496 91 : int done = 0;
1497 91 : if (typ(target) != t_VEC && typ(target) != t_COL ) pari_err_TYPE("cvp0",target);
1498 91 : if (n != nt) pari_err_TYPE("cvp0 [different dimensions]",target);
1499 77 : if (!BORNE)
1500 0 : sBORNE = 0.;
1501 : else
1502 : {
1503 77 : if (typ(BORNE) != t_REAL && typ(BORNE) != t_INT && typ(BORNE) != t_FRAC ) pari_err_TYPE("cvp0",BORNE);
1504 77 : sBORNE = gtodouble(BORNE); set_avma(av);
1505 77 : if (sBORNE < 0.) sBORNE = 0.;
1506 : }
1507 77 : if (!STOCKMAX)
1508 : {
1509 77 : stockall = 1;
1510 77 : maxrank = 200;
1511 : }
1512 : else
1513 : {
1514 0 : STOCKMAX = gfloor(STOCKMAX);
1515 0 : if (typ(STOCKMAX) != t_INT) pari_err_TYPE("cvp0",STOCKMAX);
1516 0 : maxrank = itos(STOCKMAX);
1517 0 : if (maxrank < 0)
1518 0 : pari_err_TYPE("cvp0 [negative number of vectors]",STOCKMAX);
1519 : }
1520 :
1521 77 : L = (flag==min_ALL) ? new_chunk(1+maxrank) : NULL;
1522 77 : if (n == 0 ) {
1523 7 : if (flag==min_ALL) {
1524 7 : retmkvec3(gen_0, gen_0, cgetg(1, t_MAT));
1525 : }
1526 : else {
1527 0 : return cgetg(1,t_VEC);
1528 : }
1529 : }
1530 :
1531 70 : cvp_alloc(n+1, &q, &x, &y, &z, &v, &tt, &tpre);
1532 :
1533 70 : forqfvec_init_dolll(&qv, &a, dolll);
1534 70 : av1 = avma;
1535 70 : r = qv.r;
1536 70 : u = qv.u;
1537 392 : for (j=1; j<=n; j++)
1538 : {
1539 322 : v[j] = rtodbl(gcoeff(r,j,j));
1540 1729 : for (i=1; i<j; i++) q[i][j] = rtodbl(gcoeff(r,i,j)); /* |.| <= 1/2 */
1541 : }
1542 :
1543 70 : if( dolll ) {
1544 : /* compute U^-1 * tt */
1545 70 : uinv = ZM_inv(u, &pd);
1546 70 : tv = RgM_RgC_mul(uinv, target);
1547 392 : for (j=1; j<=n; j++)
1548 : {
1549 322 : tt[j] = gtodouble(gel(tv, j));
1550 : }
1551 : } else {
1552 0 : for (j=1; j<=n; j++)
1553 : {
1554 0 : tt[j] = gtodouble(gel(target, j));
1555 : }
1556 : }
1557 :
1558 70 : if (sBORNE) maxnorm = 0.;
1559 : else
1560 : {
1561 28 : GEN B = gcoeff(a,1,1);
1562 112 : for (i = 2; i <= n; i++)
1563 84 : B = addii(B, gcoeff(a,i,i));
1564 28 : maxnorm = -1.; /* don't update maxnorm */
1565 28 : if (is_bigint(B)) return NULL;
1566 28 : sBORNE = 0.;
1567 140 : for(i=1; i<=n; i++)
1568 112 : sBORNE += v[i];
1569 : }
1570 70 : BOUND = sBORNE * (1 + eps);
1571 :
1572 : /* precompute contribution of tt to z[l] */
1573 :
1574 392 : for(k=1; k <= n; k++) {
1575 322 : tpre[k] = -tt[k];
1576 1729 : for(j=k+1; j<=n; j++) {
1577 1407 : tpre[k] -= q[k][j] * tt[j];
1578 : }
1579 : }
1580 :
1581 70 : s = 0;
1582 70 : k = n; y[n] = 0;
1583 70 : z[n] = tpre[n];
1584 70 : x[n] = (long)floor(sqrt(BOUND/v[n])-z[n]);
1585 889 : for(;;x[1]--)
1586 : {
1587 : do
1588 : {
1589 8582 : if (k>1)
1590 : {
1591 7665 : long l = k-1;
1592 7665 : z[l] = tpre[l];
1593 61488 : for (j=k; j<=n; j++) z[l] += q[l][j]*x[j];
1594 7665 : p = (double)x[k] + z[k];
1595 7665 : y[l] = y[k] + p*p*v[k];
1596 7665 : x[l] = (long)floor(sqrt((BOUND-y[l])/v[l])-z[l]);
1597 7665 : k = l;
1598 : }
1599 : for(;;)
1600 : {
1601 16247 : p = (double)x[k] + z[k];
1602 16247 : if (y[k] + p*p*v[k] <= BOUND) break;
1603 7735 : if (k >= n) {
1604 70 : done = 1;
1605 70 : break;
1606 : }
1607 7665 : k++; x[k]--;
1608 : }
1609 : }
1610 8582 : while (k > 1 && !done);
1611 959 : if (done) break;
1612 :
1613 889 : p = (double)x[1] + z[1];
1614 889 : p = y[1] + p*p*v[1]; /* norm(x-target) */
1615 889 : if (maxnorm >= 0)
1616 : {
1617 175 : if (p > maxnorm) maxnorm = p;
1618 : }
1619 : else
1620 : { /* maxnorm < 0 : only look for closest vectors */
1621 714 : if (p * (1+10*eps) < sBORNE) {
1622 231 : sBORNE = p; set_avma(av1);
1623 231 : BOUND = sBORNE * (1+eps);
1624 231 : L = new_chunk(maxrank+1);
1625 231 : s = 0;
1626 : }
1627 : }
1628 889 : s++;
1629 :
1630 889 : switch(flag)
1631 : {
1632 0 : case min_FIRST:
1633 0 : if (dolll) x = ZM_zc_mul(u,x);
1634 0 : return gerepilecopy(av, mkvec2(dbltor(p), x));
1635 :
1636 889 : case min_ALL:
1637 889 : if (s > maxrank && stockall) /* overflow */
1638 : {
1639 0 : long maxranknew = maxrank << 1;
1640 0 : GEN Lnew = new_chunk(1 + maxranknew);
1641 0 : for (i=1; i<=maxrank; i++) Lnew[i] = L[i];
1642 0 : L = Lnew; maxrank = maxranknew;
1643 : }
1644 889 : if (s<=maxrank) gel(L,s) = leafcopy(x);
1645 889 : break;
1646 : }
1647 : }
1648 70 : switch(flag)
1649 : {
1650 0 : case min_FIRST:
1651 0 : set_avma(av); return cgetg(1,t_VEC);
1652 : }
1653 70 : r = (maxnorm >= 0) ? dbltor(maxnorm): dbltor(sBORNE);
1654 70 : k = minss(s,maxrank);
1655 70 : L[0] = evaltyp(t_MAT) | evallg(k + 1);
1656 322 : for (j=1; j<=k; j++)
1657 252 : gel(L,j) = (dolll==1) ? ZM_zc_mul(u, gel(L,j)) : zc_to_ZC(gel(L,j));
1658 70 : return gerepilecopy(av, mkvec3(stoi(s), r, L));
1659 : }
1660 :
1661 : static GEN
1662 553 : minim0(GEN a, GEN BORNE, GEN STOCKMAX, long flag)
1663 : {
1664 553 : GEN v = minim0_dolll(a, BORNE, STOCKMAX, flag, 1);
1665 552 : if (!v) pari_err_PREC("qfminim");
1666 546 : return v;
1667 : }
1668 :
1669 : static GEN
1670 91 : cvp0(GEN a, GEN target, GEN BORNE, GEN STOCKMAX, long flag)
1671 : {
1672 91 : GEN v = cvp0_dolll(a, target, BORNE, STOCKMAX, flag, 1);
1673 77 : if (!v) pari_err_PREC("qfcvp");
1674 77 : return v;
1675 : }
1676 :
1677 : static GEN
1678 252 : minim0_zm(GEN a, GEN BORNE, GEN STOCKMAX, long flag)
1679 : {
1680 252 : GEN v = minim0_dolll(a, BORNE, STOCKMAX, flag, 2);
1681 252 : if (!v) pari_err_PREC("qfminim");
1682 252 : return v;
1683 : }
1684 :
1685 : GEN
1686 462 : qfrep0(GEN a, GEN borne, long flag)
1687 462 : { return minim0(a, borne, gen_0, (flag & 1)? min_VECSMALL2: min_VECSMALL); }
1688 :
1689 : GEN
1690 133 : qfminim0(GEN a, GEN borne, GEN stockmax, long flag, long prec)
1691 : {
1692 133 : switch(flag)
1693 : {
1694 49 : case 0: return minim0(a,borne,stockmax,min_ALL);
1695 35 : case 1: return minim0(a,borne,gen_0 ,min_FIRST);
1696 49 : case 2:
1697 : {
1698 49 : long maxnum = -1;
1699 49 : if (typ(a) != t_MAT) pari_err_TYPE("qfminim",a);
1700 49 : if (stockmax) {
1701 14 : if (typ(stockmax) != t_INT) pari_err_TYPE("qfminim",stockmax);
1702 14 : maxnum = itos(stockmax);
1703 : }
1704 49 : a = fincke_pohst(a,borne,maxnum,prec,NULL);
1705 42 : if (!a) pari_err_PREC("qfminim");
1706 42 : return a;
1707 : }
1708 0 : default: pari_err_FLAG("qfminim");
1709 : }
1710 : return NULL; /* LCOV_EXCL_LINE */
1711 : }
1712 :
1713 :
1714 : GEN
1715 91 : qfcvp0(GEN a, GEN target, GEN borne, GEN stockmax, long flag)
1716 : {
1717 91 : switch(flag)
1718 : {
1719 91 : case 0: return cvp0(a,target,borne,stockmax,min_ALL);
1720 0 : case 1: return cvp0(a,target,borne,gen_0 ,min_FIRST);
1721 : /* case 2:
1722 : TODO: more robust finke_pohst enumeration */
1723 0 : default: pari_err_FLAG("qfcvp");
1724 : }
1725 : return NULL; /* LCOV_EXCL_LINE */
1726 : }
1727 :
1728 : GEN
1729 7 : minim(GEN a, GEN borne, GEN stockmax)
1730 7 : { return minim0(a,borne,stockmax,min_ALL); }
1731 :
1732 : GEN
1733 252 : minim_zm(GEN a, GEN borne, GEN stockmax)
1734 252 : { return minim0_zm(a,borne,stockmax,min_ALL); }
1735 :
1736 : GEN
1737 42 : minim_raw(GEN a, GEN BORNE, GEN STOCKMAX)
1738 42 : { return minim0_dolll(a, BORNE, STOCKMAX, min_ALL, 0); }
1739 :
1740 : GEN
1741 0 : minim2(GEN a, GEN borne, GEN stockmax)
1742 0 : { return minim0(a,borne,stockmax,min_FIRST); }
1743 :
1744 : /* If V depends linearly from the columns of the matrix, return 0.
1745 : * Otherwise, update INVP and L and return 1. No GC. */
1746 : static int
1747 1652 : addcolumntomatrix(GEN V, GEN invp, GEN L)
1748 : {
1749 1652 : long i,j,k, n = lg(invp);
1750 1652 : GEN a = cgetg(n, t_COL), ak = NULL, mak;
1751 :
1752 84231 : for (k = 1; k < n; k++)
1753 83706 : if (!L[k])
1754 : {
1755 27902 : ak = RgMrow_zc_mul(invp, V, k);
1756 27902 : if (!gequal0(ak)) break;
1757 : }
1758 1652 : if (k == n) return 0;
1759 1127 : L[k] = 1;
1760 1127 : mak = gneg_i(ak);
1761 43253 : for (i=k+1; i<n; i++)
1762 42126 : gel(a,i) = gdiv(RgMrow_zc_mul(invp, V, i), mak);
1763 43883 : for (j=1; j<=k; j++)
1764 : {
1765 42756 : GEN c = gel(invp,j), ck = gel(c,k);
1766 42756 : if (gequal0(ck)) continue;
1767 8757 : gel(c,k) = gdiv(ck, ak);
1768 8757 : if (j==k)
1769 43253 : for (i=k+1; i<n; i++)
1770 42126 : gel(c,i) = gmul(gel(a,i), ck);
1771 : else
1772 184814 : for (i=k+1; i<n; i++)
1773 177184 : gel(c,i) = gadd(gel(c,i), gmul(gel(a,i), ck));
1774 : }
1775 1127 : return 1;
1776 : }
1777 :
1778 : GEN
1779 42 : qfperfection(GEN a)
1780 : {
1781 42 : pari_sp av = avma;
1782 : GEN u, L;
1783 42 : long r, s, k, l, n = lg(a)-1;
1784 :
1785 42 : if (!n) return gen_0;
1786 42 : if (typ(a) != t_MAT || !RgM_is_ZM(a)) pari_err_TYPE("qfperfection",a);
1787 42 : a = minim_lll(a, &u);
1788 42 : L = minim_raw(a,NULL,NULL);
1789 42 : r = (n*(n+1)) >> 1;
1790 42 : if (L)
1791 : {
1792 : GEN D, V, invp;
1793 35 : L = gel(L, 3); l = lg(L);
1794 35 : if (l == 2) { set_avma(av); return gen_1; }
1795 : /* |L[i]|^2 fits into a long for all i */
1796 21 : D = zero_zv(r);
1797 21 : V = cgetg(r+1, t_VECSMALL);
1798 21 : invp = matid(r);
1799 21 : s = 0;
1800 1659 : for (k = 1; k < l; k++)
1801 : {
1802 1652 : pari_sp av2 = avma;
1803 1652 : GEN x = gel(L,k);
1804 : long i, j, I;
1805 21098 : for (i = I = 1; i<=n; i++)
1806 145278 : for (j=i; j<=n; j++,I++) V[I] = x[i]*x[j];
1807 1652 : if (!addcolumntomatrix(V,invp,D)) set_avma(av2);
1808 1127 : else if (++s == r) break;
1809 : }
1810 : }
1811 : else
1812 : {
1813 : GEN M;
1814 7 : L = fincke_pohst(a,NULL,-1, DEFAULTPREC, NULL);
1815 7 : if (!L) pari_err_PREC("qfminim");
1816 7 : L = gel(L, 3); l = lg(L);
1817 7 : if (l == 2) { set_avma(av); return gen_1; }
1818 7 : M = cgetg(l, t_MAT);
1819 959 : for (k = 1; k < l; k++)
1820 : {
1821 952 : GEN x = gel(L,k), c = cgetg(r+1, t_COL);
1822 : long i, I, j;
1823 16184 : for (i = I = 1; i<=n; i++)
1824 144704 : for (j=i; j<=n; j++,I++) gel(c,I) = mulii(gel(x,i), gel(x,j));
1825 952 : gel(M,k) = c;
1826 : }
1827 7 : s = ZM_rank(M);
1828 : }
1829 28 : return gc_utoipos(av, s);
1830 : }
1831 :
1832 : static GEN
1833 141 : clonefill(GEN S, long s, long t)
1834 : { /* initialize to dummy values */
1835 141 : GEN T = S, dummy = cgetg(1, t_STR);
1836 : long i;
1837 310917 : for (i = s+1; i <= t; i++) gel(S,i) = dummy;
1838 141 : S = gclone(S); if (isclone(T)) gunclone(T);
1839 141 : return S;
1840 : }
1841 :
1842 : /* increment ZV x, by incrementing cell of index k. Initial value x0[k] was
1843 : * chosen to minimize qf(x) for given x0[1..k-1] and x0[k+1,..] = 0
1844 : * The last nonzero entry must be positive and goes through x0[k]+1,2,3,...
1845 : * Others entries go through: x0[k]+1,-1,2,-2,...*/
1846 : INLINE void
1847 2952133 : step(GEN x, GEN y, GEN inc, long k)
1848 : {
1849 2952133 : if (!signe(gel(y,k))) /* x[k+1..] = 0 */
1850 160647 : gel(x,k) = addiu(gel(x,k), 1); /* leading coeff > 0 */
1851 : else
1852 : {
1853 2791486 : long i = inc[k];
1854 2791486 : gel(x,k) = addis(gel(x,k), i),
1855 2791483 : inc[k] = (i > 0)? -1-i: 1-i;
1856 : }
1857 2952130 : }
1858 :
1859 : /* 1 if we are "sure" that x < y, up to few rounding errors, i.e.
1860 : * x < y - epsilon. More precisely :
1861 : * - sign(x - y) < 0
1862 : * - lgprec(x-y) > 3 || expo(x - y) - expo(x) > -24 */
1863 : static int
1864 1216163 : mplessthan(GEN x, GEN y)
1865 : {
1866 1216163 : pari_sp av = avma;
1867 1216163 : GEN z = mpsub(x, y);
1868 1216162 : set_avma(av);
1869 1216162 : if (typ(z) == t_INT) return (signe(z) < 0);
1870 1216162 : if (signe(z) >= 0) return 0;
1871 22171 : if (realprec(z) > LOWDEFAULTPREC) return 1;
1872 22171 : return ( expo(z) - mpexpo(x) > -24 );
1873 : }
1874 :
1875 : /* 1 if we are "sure" that x > y, up to few rounding errors, i.e.
1876 : * x > y + epsilon */
1877 : static int
1878 4620407 : mpgreaterthan(GEN x, GEN y)
1879 : {
1880 4620407 : pari_sp av = avma;
1881 4620407 : GEN z = mpsub(x, y);
1882 4620431 : set_avma(av);
1883 4620454 : if (typ(z) == t_INT) return (signe(z) > 0);
1884 4620454 : if (signe(z) <= 0) return 0;
1885 2689298 : if (realprec(z) > LOWDEFAULTPREC) return 1;
1886 475821 : return ( expo(z) - mpexpo(x) > -24 );
1887 : }
1888 :
1889 : /* x a t_INT, y t_INT or t_REAL */
1890 : INLINE GEN
1891 1228236 : mulimp(GEN x, GEN y)
1892 : {
1893 1228236 : if (typ(y) == t_INT) return mulii(x,y);
1894 1228236 : return signe(x) ? mulir(x,y): gen_0;
1895 : }
1896 : /* x + y*z, x,z two mp's, y a t_INT */
1897 : INLINE GEN
1898 13537668 : addmulimp(GEN x, GEN y, GEN z)
1899 : {
1900 13537668 : if (!signe(y)) return x;
1901 5830596 : if (typ(z) == t_INT) return mpadd(x, mulii(y, z));
1902 5830596 : return mpadd(x, mulir(y, z));
1903 : }
1904 :
1905 : /* yk + vk * (xk + zk)^2 */
1906 : static GEN
1907 5779080 : norm_aux(GEN xk, GEN yk, GEN zk, GEN vk)
1908 : {
1909 5779080 : GEN t = mpadd(xk, zk);
1910 5779070 : if (typ(t) == t_INT) { /* probably gen_0, avoid loss of accuracy */
1911 305900 : yk = addmulimp(yk, sqri(t), vk);
1912 : } else {
1913 5473170 : yk = mpadd(yk, mpmul(sqrr(t), vk));
1914 : }
1915 5779046 : return yk;
1916 : }
1917 : /* yk + vk * (xk + zk)^2 < B + epsilon */
1918 : static int
1919 4166388 : check_bound(GEN B, GEN xk, GEN yk, GEN zk, GEN vk)
1920 : {
1921 4166388 : pari_sp av = avma;
1922 4166388 : int f = mpgreaterthan(norm_aux(xk,yk,zk,vk), B);
1923 4166387 : return gc_bool(av, !f);
1924 : }
1925 :
1926 : /* q(k-th canonical basis vector), where q is given in Cholesky form
1927 : * q(x) = sum_{i = 1}^n q[i,i] (x[i] + sum_{j > i} q[i,j] x[j])^2.
1928 : * Namely q(e_k) = q[k,k] + sum_{i < k} q[i,i] q[i,k]^2
1929 : * Assume 1 <= k <= n. */
1930 : static GEN
1931 182 : cholesky_norm_ek(GEN q, long k)
1932 : {
1933 182 : GEN t = gcoeff(q,k,k);
1934 : long i;
1935 1484 : for (i = 1; i < k; i++) t = norm_aux(gen_0, t, gcoeff(q,i,k), gcoeff(q,i,i));
1936 182 : return t;
1937 : }
1938 :
1939 : /* q is the Cholesky decomposition of a quadratic form
1940 : * Enumerate vectors whose norm is less than BORNE (Algo 2.5.7),
1941 : * minimal vectors if BORNE = NULL (implies check = NULL).
1942 : * If (check != NULL) consider only vectors passing the check, and assumes
1943 : * we only want the smallest possible vectors */
1944 : static GEN
1945 14692 : smallvectors(GEN q, GEN BORNE, long maxnum, FP_chk_fun *CHECK)
1946 : {
1947 14692 : long N = lg(q), n = N-1, i, j, k, s, stockmax, checkcnt = 1;
1948 : pari_sp av, av1;
1949 : GEN inc, S, x, y, z, v, p1, alpha, norms;
1950 : GEN norme1, normax1, borne1, borne2;
1951 14692 : GEN (*check)(void *,GEN) = CHECK? CHECK->f: NULL;
1952 14692 : void *data = CHECK? CHECK->data: NULL;
1953 14692 : const long skipfirst = CHECK? CHECK->skipfirst: 0;
1954 14692 : const int stockall = (maxnum == -1);
1955 :
1956 14692 : alpha = dbltor(0.95);
1957 14692 : normax1 = gen_0;
1958 :
1959 14692 : v = cgetg(N,t_VEC);
1960 14692 : inc = const_vecsmall(n, 1);
1961 :
1962 14692 : av = avma;
1963 14692 : stockmax = stockall? 2000: maxnum;
1964 14692 : norms = cgetg(check?(stockmax+1): 1,t_VEC); /* unused if (!check) */
1965 14692 : S = cgetg(stockmax+1,t_VEC);
1966 14692 : x = cgetg(N,t_COL);
1967 14692 : y = cgetg(N,t_COL);
1968 14692 : z = cgetg(N,t_COL);
1969 97716 : for (i=1; i<N; i++) {
1970 83024 : gel(v,i) = gcoeff(q,i,i);
1971 83024 : gel(x,i) = gel(y,i) = gel(z,i) = gen_0;
1972 : }
1973 14692 : if (BORNE)
1974 : {
1975 14671 : borne1 = BORNE;
1976 14671 : if (gsigne(borne1) <= 0) retmkvec3(gen_0, gen_0, cgetg(1,t_MAT));
1977 14657 : if (typ(borne1) != t_REAL)
1978 : {
1979 : long prec;
1980 419 : prec = nbits2prec(gexpo(borne1) + 10);
1981 419 : borne1 = gtofp(borne1, maxss(prec, DEFAULTPREC));
1982 : }
1983 : }
1984 : else
1985 : {
1986 21 : borne1 = gcoeff(q,1,1);
1987 203 : for (i=2; i<N; i++)
1988 : {
1989 182 : GEN b = cholesky_norm_ek(q, i);
1990 182 : if (gcmp(b, borne1) < 0) borne1 = b;
1991 : }
1992 : /* borne1 = norm of smallest basis vector */
1993 : }
1994 14678 : borne2 = mulrr(borne1,alpha);
1995 14678 : if (DEBUGLEVEL>2)
1996 0 : err_printf("smallvectors looking for norm < %P.4G\n",borne1);
1997 14678 : s = 0; k = n;
1998 383899 : for(;; step(x,y,inc,k)) /* main */
1999 : { /* x (supposedly) small vector, ZV.
2000 : * For all t >= k, we have
2001 : * z[t] = sum_{j > t} q[t,j] * x[j]
2002 : * y[t] = sum_{i > t} q[i,i] * (x[i] + z[i])^2
2003 : * = 0 <=> x[i]=0 for all i>t */
2004 : do
2005 : {
2006 1612133 : int skip = 0;
2007 1612133 : if (k > 1)
2008 : {
2009 1228236 : long l = k-1;
2010 1228236 : av1 = avma;
2011 1228236 : p1 = mulimp(gel(x,k), gcoeff(q,l,k));
2012 14460011 : for (j=k+1; j<N; j++) p1 = addmulimp(p1, gel(x,j), gcoeff(q,l,j));
2013 1228235 : gel(z,l) = gerepileuptoleaf(av1,p1);
2014 :
2015 1228235 : av1 = avma;
2016 1228235 : p1 = norm_aux(gel(x,k), gel(y,k), gel(z,k), gel(v,k));
2017 1228233 : gel(y,l) = gerepileuptoleaf(av1, p1);
2018 : /* skip the [x_1,...,x_skipfirst,0,...,0] */
2019 1228236 : if ((l <= skipfirst && !signe(gel(y,skipfirst)))
2020 1228236 : || mplessthan(borne1, gel(y,l))) skip = 1;
2021 : else /* initial value, minimizing (x[l] + z[l])^2, hence qf(x) for
2022 : the given x[1..l-1] */
2023 1214272 : gel(x,l) = mpround( mpneg(gel(z,l)) );
2024 1228234 : k = l;
2025 : }
2026 1228235 : for(;; step(x,y,inc,k))
2027 : { /* at most 2n loops */
2028 2840364 : if (!skip)
2029 : {
2030 2826401 : if (check_bound(borne1, gel(x,k),gel(y,k),gel(z,k),gel(v,k))) break;
2031 1339990 : step(x,y,inc,k);
2032 1340000 : if (check_bound(borne1, gel(x,k),gel(y,k),gel(z,k),gel(v,k))) break;
2033 : }
2034 1242913 : skip = 0; inc[k] = 1;
2035 1242913 : if (++k > n) goto END;
2036 : }
2037 :
2038 1597463 : if (gc_needed(av,2))
2039 : {
2040 15 : if(DEBUGMEM>1) pari_warn(warnmem,"smallvectors");
2041 15 : if (stockmax) S = clonefill(S, s, stockmax);
2042 15 : if (check) {
2043 15 : GEN dummy = cgetg(1, t_STR);
2044 9629 : for (i=s+1; i<=stockmax; i++) gel(norms,i) = dummy;
2045 : }
2046 15 : gerepileall(av,7,&x,&y,&z,&normax1,&borne1,&borne2,&norms);
2047 : }
2048 : }
2049 1597463 : while (k > 1);
2050 383899 : if (!signe(gel(x,1)) && !signe(gel(y,1))) continue; /* exclude 0 */
2051 :
2052 383184 : av1 = avma;
2053 383184 : norme1 = norm_aux(gel(x,1),gel(y,1),gel(z,1),gel(v,1));
2054 383184 : if (mpgreaterthan(norme1,borne1)) { set_avma(av1); continue; /* main */ }
2055 :
2056 383184 : norme1 = gerepileuptoleaf(av1,norme1);
2057 383184 : if (check)
2058 : {
2059 314598 : if (checkcnt < 5 && mpcmp(norme1, borne2) < 0)
2060 : {
2061 4381 : if (!check(data,x)) { checkcnt++ ; continue; /* main */}
2062 481 : if (DEBUGLEVEL>4) err_printf("New bound: %Ps", norme1);
2063 481 : borne1 = norme1;
2064 481 : borne2 = mulrr(borne1, alpha);
2065 481 : s = 0; checkcnt = 0;
2066 : }
2067 : }
2068 : else
2069 : {
2070 68586 : if (!BORNE) /* find minimal vectors */
2071 : {
2072 1890 : if (mplessthan(norme1, borne1))
2073 : { /* strictly smaller vector than previously known */
2074 0 : borne1 = norme1; /* + epsilon */
2075 0 : s = 0;
2076 : }
2077 : }
2078 : else
2079 66696 : if (mpcmp(norme1,normax1) > 0) normax1 = norme1;
2080 : }
2081 379284 : if (++s > stockmax) continue; /* too many vectors: no longer remember */
2082 378353 : if (check) gel(norms,s) = norme1;
2083 378353 : gel(S,s) = leafcopy(x);
2084 378353 : if (s != stockmax) continue; /* still room, get next vector */
2085 :
2086 126 : if (check)
2087 : { /* overflow, eliminate vectors failing "check" */
2088 105 : pari_sp av2 = avma;
2089 : long imin, imax;
2090 105 : GEN per = indexsort(norms), S2 = cgetg(stockmax+1, t_VEC);
2091 105 : if (DEBUGLEVEL>2) err_printf("sorting... [%ld elts]\n",s);
2092 : /* let N be the minimal norm so far for x satisfying 'check'. Keep
2093 : * all elements of norm N */
2094 26593 : for (i = 1; i <= s; i++)
2095 : {
2096 26586 : long k = per[i];
2097 26586 : if (check(data,gel(S,k))) { borne1 = gel(norms,k); break; }
2098 : }
2099 105 : imin = i;
2100 20943 : for (; i <= s; i++)
2101 20922 : if (mpgreaterthan(gel(norms,per[i]), borne1)) break;
2102 105 : imax = i;
2103 20943 : for (i=imin, s=0; i < imax; i++) gel(S2,++s) = gel(S,per[i]);
2104 20943 : for (i = 1; i <= s; i++) gel(S,i) = gel(S2,i);
2105 105 : set_avma(av2);
2106 105 : if (s) { borne2 = mulrr(borne1, alpha); checkcnt = 0; }
2107 105 : if (!stockall) continue;
2108 105 : if (s > stockmax/2) stockmax <<= 1;
2109 105 : norms = cgetg(stockmax+1, t_VEC);
2110 20943 : for (i = 1; i <= s; i++) gel(norms,i) = borne1;
2111 : }
2112 : else
2113 : {
2114 21 : if (!stockall && BORNE) goto END;
2115 21 : if (!stockall) continue;
2116 21 : stockmax <<= 1;
2117 : }
2118 :
2119 : {
2120 126 : GEN Snew = clonefill(vec_lengthen(S,stockmax), s, stockmax);
2121 126 : if (isclone(S)) gunclone(S);
2122 126 : S = Snew;
2123 : }
2124 : }
2125 14678 : END:
2126 14678 : if (s < stockmax) stockmax = s;
2127 14678 : if (check)
2128 : {
2129 : GEN per, alph, pols, p;
2130 14650 : if (DEBUGLEVEL>2) err_printf("final sort & check...\n");
2131 14650 : setlg(norms,stockmax+1); per = indexsort(norms);
2132 14650 : alph = cgetg(stockmax+1,t_VEC);
2133 14650 : pols = cgetg(stockmax+1,t_VEC);
2134 84457 : for (j=0,i=1; i<=stockmax; i++)
2135 : {
2136 70058 : long t = per[i];
2137 70058 : GEN N = gel(norms,t);
2138 70058 : if (j && mpgreaterthan(N, borne1)) break;
2139 69807 : if ((p = check(data,gel(S,t))))
2140 : {
2141 55866 : if (!j) borne1 = N;
2142 55866 : j++;
2143 55866 : gel(pols,j) = p;
2144 55866 : gel(alph,j) = gel(S,t);
2145 : }
2146 : }
2147 14650 : setlg(pols,j+1);
2148 14650 : setlg(alph,j+1);
2149 14650 : if (stockmax && isclone(S)) { alph = gcopy(alph); gunclone(S); }
2150 14650 : return mkvec2(pols, alph);
2151 : }
2152 28 : if (stockmax)
2153 : {
2154 21 : setlg(S,stockmax+1);
2155 21 : settyp(S,t_MAT);
2156 21 : if (isclone(S)) { p1 = S; S = gcopy(S); gunclone(p1); }
2157 : }
2158 : else
2159 7 : S = cgetg(1,t_MAT);
2160 28 : return mkvec3(utoi(s<<1), borne1, S);
2161 : }
2162 :
2163 : /* solve q(x) = x~.a.x <= bound, a > 0.
2164 : * If check is non-NULL keep x only if check(x).
2165 : * If a is a vector, assume a[1] is the LLL-reduced Cholesky form of q */
2166 : GEN
2167 14713 : fincke_pohst(GEN a, GEN B0, long stockmax, long PREC, FP_chk_fun *CHECK)
2168 : {
2169 14713 : pari_sp av = avma;
2170 : VOLATILE long i,j,l;
2171 14713 : VOLATILE GEN r,rinv,rinvtrans,u,v,res,z,vnorm,rperm,perm,uperm, bound = B0;
2172 :
2173 14713 : if (typ(a) == t_VEC)
2174 : {
2175 14245 : r = gel(a,1);
2176 14245 : u = NULL;
2177 : }
2178 : else
2179 : {
2180 468 : long prec = PREC;
2181 468 : l = lg(a);
2182 468 : if (l == 1)
2183 : {
2184 7 : if (CHECK) pari_err_TYPE("fincke_pohst [dimension 0]", a);
2185 7 : retmkvec3(gen_0, gen_0, cgetg(1,t_MAT));
2186 : }
2187 461 : u = lllfp(a, 0.75, LLL_GRAM | LLL_IM);
2188 454 : if (!u || lg(u) != lg(a)) return gc_NULL(av);
2189 454 : r = qf_RgM_apply(a,u);
2190 454 : i = gprecision(r);
2191 454 : if (i)
2192 412 : prec = i;
2193 : else {
2194 42 : prec = DEFAULTPREC + nbits2extraprec(gexpo(r));
2195 42 : if (prec < PREC) prec = PREC;
2196 : }
2197 454 : if (DEBUGLEVEL>2) err_printf("first LLL: prec = %ld\n", prec);
2198 454 : r = qfgaussred_positive(r);
2199 454 : if (!r) return gc_NULL(av);
2200 1984 : for (i=1; i<l; i++)
2201 : {
2202 1530 : GEN s = gsqrt(gcoeff(r,i,i), prec);
2203 1530 : gcoeff(r,i,i) = s;
2204 4236 : for (j=i+1; j<l; j++) gcoeff(r,i,j) = gmul(s, gcoeff(r,i,j));
2205 : }
2206 : }
2207 : /* now r~ * r = a in LLL basis */
2208 14699 : rinv = RgM_inv_upper(r);
2209 14699 : if (!rinv) return gc_NULL(av);
2210 14699 : rinvtrans = shallowtrans(rinv);
2211 14699 : if (DEBUGLEVEL>2)
2212 0 : err_printf("Fincke-Pohst, final LLL: prec = %ld\n", gprecision(rinvtrans));
2213 14699 : v = lll(rinvtrans);
2214 14699 : if (lg(v) != lg(rinvtrans)) return gc_NULL(av);
2215 :
2216 14699 : rinvtrans = RgM_mul(rinvtrans, v);
2217 14699 : v = ZM_inv(shallowtrans(v),NULL);
2218 14699 : r = RgM_mul(r,v);
2219 14699 : u = u? ZM_mul(u,v): v;
2220 :
2221 14699 : l = lg(r);
2222 14699 : vnorm = cgetg(l,t_VEC);
2223 97751 : for (j=1; j<l; j++) gel(vnorm,j) = gnorml2(gel(rinvtrans,j));
2224 14699 : rperm = cgetg(l,t_MAT);
2225 14699 : uperm = cgetg(l,t_MAT); perm = indexsort(vnorm);
2226 97751 : for (i=1; i<l; i++) { uperm[l-i] = u[perm[i]]; rperm[l-i] = r[perm[i]]; }
2227 14699 : u = uperm;
2228 14699 : r = rperm; res = NULL;
2229 14699 : pari_CATCH(e_PREC) { }
2230 : pari_TRY {
2231 : GEN q;
2232 14699 : if (CHECK && CHECK->f_init) bound = CHECK->f_init(CHECK, r, u);
2233 14692 : q = gaussred_from_QR(r, gprecision(vnorm));
2234 14692 : if (q) res = smallvectors(q, bound, stockmax, CHECK);
2235 14692 : } pari_ENDCATCH;
2236 14699 : if (!res) return gc_NULL(av);
2237 14692 : if (CHECK)
2238 : {
2239 14650 : if (CHECK->f_post) res = CHECK->f_post(CHECK, res, u);
2240 14650 : return res;
2241 : }
2242 :
2243 42 : z = cgetg(4,t_VEC);
2244 42 : gel(z,1) = gcopy(gel(res,1));
2245 42 : gel(z,2) = gcopy(gel(res,2));
2246 42 : gel(z,3) = ZM_mul(u, gel(res,3)); return gerepileupto(av,z);
2247 : }
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